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Mathematics-1
Gujarat Technological University 2018
About the Authors
Ravish R Singh is presently Director at Thakur Ramnarayan College

of Arts and Commerce, Mumbai. He obtained a BE degree from
University of Mumbai in 1991, an MTech degree from IIT Bombay
in 2001, and a PhD degree from Faculty of Technology, University
of Mumbai, in 2013. He has published several books with McGraw
Hill Education (India) on varied subjects like Engineering
Mathematics, Applied Mathematics, Electrical Networks, Network
Analysis and Synthesis, Electrical Engineering, Basic Electrical and
Electronics Engineering, etc., for all-India curricula as well as regional curricula of
some universities like Gujarat Technological University, Mumbai University, Pune
University, Jawaharlal Nehru Technological University, Anna University, Uttarakhand
Technical University, and Dr A P J Abdul Kalam Technical University. Dr Singh is a
member of IEEE, ISTE, and IETE, and has published research papers in national and
international journals. His fields of interest include Circuits, Signals and Systems, and
Engineering Mathematics.
Mukul Bhatt is presently Assistant Professor at Department of

Mathematics and Statistics, Thakur Ramnarayan College of Arts and
Commerce, Mumbai. She obtained her MSc (Mathematics) degree
from H N B Garhwal University in 1992, and a PhD degree from
Faculty of Science, PAHER University, Udaipur, Rajasthan in 2017.
She has published several books with McGraw Hill Education (India)
Private Limited on Engineering Mathematics and Applied Mathematics
for all-India curricula as well as regional curricula of some universities like Gujarat
Technological University, Mumbai University, Pune University, Jawaharlal Nehru
Technological University, Anna University, Uttarakhand Technical University, and
Dr A P J Abdul Kalam Technical University. Dr Bhatt has twenty six years of teaching
experience at various levels in engineering colleges and her fields of interest include
Integral Calculus, Complex Analysis, and Operation Research. She is a member of
ISTE.
Mathematics-1
Gujarat Technological University 2018
Ravish R Singh
Director
Thakur Ramnarayan College of Arts & Commerce
Mumbai, Maharashtra
Mukul Bhatt
Assistant Professor
Thakur Ramnarayan College of Arts & Commerce
Mumbai, Maharashtra
McGraw Hill Education (India) Private Limited

Chennai
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Dedicated
to
Aman and Aditri
Ravish R Singh
Soumya and Siddharth

Mukul Bhatt
Contents
Preface xi
Roadmap to the Syllabus xv
UNIT-1
1. Indeterminate Forms 1.1–1.63
1.1 Introduction 1.1
1.2 L’Hospital’s Rule 1.1
0
1.3 Type 1: Form 1.2
0
•
1.4 Type 2: form 1.16
•
1.5 Type 3: 0 × • Form 1.23
1.6 Type 4: • − • Form 1.30
1.7 Type 5: 1•, • 0, 0 0 Forms 1.38
Points to Remember 1.60
Multiple Choice Questions 1.60
2. Improper Integrals 2.1–2.25
2.2 Improper Integrals 2.1
2.3 Improper Integrals of the First Kind 2.2
2.4 Improper Integrals of the Second Kind 2.9
2.5 Improper Integral of the Third Kind 2.16
2.6 Convergence and Divergence of Improper Integrals 2.17
3. Gamma and Beta Functions 3.1–3.39
3.2 Gamma Function 3.1
3.3 Properties of Gamma Function 3.2
3.4 Beta Function 3.11
3.5 Properties of Beta Functions 3.12
3.6 Beta Function as Improper Integral 3.28
viii Contents

4. Applications of Definite Integrals 4.1–4.67
4.2 Volume Using Cross-sections 4.1
4.3 Length of Plane Curves 4.6
4.4 Area of Surface of Solid of Revolution 4.46
UNIT-2
5. Sequences and Series 5.1–5.121
5.2 Sequence 5.2
5.3 Infinite Series 5.8
5.4 The nth Term Test for Divergence 5.9
5.5 Geometric Series 5.10
5.6 Telescoping Series 5.15
5.7 Combining Series 5.18
5.8 Harmonic Series 5.19
5.9 p-Series 5.20
5.10 Comparison Test 5.20
5.11 D’Alembert’s Ratio Test 5.40
5.12 Raabe’s Test 5.67
5.13 Cauchy’s Root Test 5.73
5.14 Cauchy’s Integral Test 5.82
5.15 Alternating Series 5.87
5.16 Absolute and Conditional Convergent of a Series 5.94
5.17 Power Series 5.101
6. Taylor’s and Maclaurin’s Series 6.1–6.70
6.2 Taylor’s Series 6.1
6.3 Maclaurin’s Series 6.27
Contents ix
UNIT-3
7. Fourier Series 7.1–7.126
7.2 Periodic Functions 7.1
7.3 Orthogonality of Trigonometric System 7.2
7.4 Dirichlet’s Conditions for Representation by a Fourier Series 7.5
7.5 Trigonometric Fourier Series 7.6
7.6 Fourier Series of Functions of Period 2l 7.7
7.7 Fourier Series of Even and Odd Functions 7.66
7.8 Half-Range Fourier Series 7.93
UNIT-4
8. Partial Derivatives 8.1–8.184
8.2 Functions of Two or More Variables 8.2
8.3 Limit and Continuity of Functions of Several Variables 8.2
8.4 Partial Derivatives 8.10
8.5 Higher-Order Partial Derivatives 8.11
8.6 Total Derivatives 8.59
8.7 Implicit Differentiation 8.94
8.8 Gradient and Directional Derivative 8.103
8.9 Tangent Plane and Normal Line 8.107
8.10 Local Extreme Values (Maximum and Minimum Values) 8.116
8.11 Extreme Values with Constrained Variables 8.134
8.12 Method of Lagrange Multipliers 8.145
UNIT-5
9. Multiple Integrals 9.1–9.175
9.2 Double Integrals 9.1
9.3 Change of Order of Integration 9.31
9.4 Double Integrals in Polar Coordinates 9.66
9.5 Multiple Integrals by Substitution 9.77
9.6 Triple Integrals 9.109
x Contents
9.7 Area by Double Integrals 9.141

UNIT-6
10. Matrices 10.1–10.141
10.2 Matrix 10.2
10.3 Some Definitions Associated with Matrices 10.2
10.4 Elementary Row Operations in Matrix 10.6
10.5 Row Echelon and Reduced Row Echelon Forms of a Matrix 10.7
10.6 Rank of a Matrix 10.13
10.7 Inverse of a Matrix by Gauss–Jordan Method 10.18
10.8 System of Non-Homogeneous Linear Equations 10.22
10.9 System of Homogeneous Linear Equations 10.48
10.10 Eigenvalues and Eigenvectors 10.64
10.11 Properties of Eigenvalues 10.65
10.12 Linear Dependence and Independence of Eigenvectors 10.76
10.13 Properties of Eigenvectors 10.76
10.14 Cayley-Hamilton Theorem 10.108
10.15 Similarity Transformation 10.119
10.16 Diagonalization of a Matrix 10.119
Multiple-Choice Questions 10.137
Index I.1–I.3
Preface
Mathematics is a key area of study in any engineering course. A sound knowledge
of this subject will help engineering students develop analytical skills, and thus
enable them to solve numerical problems encountered in real life, as well as apply
mathematical principles to physical problems, particularly in the field of engineering.
Users
This book is designed for the first year GTU engineering students pursuing the course
Mathematics-1, SUBJECT CODE: 3110014 in their first year Ist Semester. It covers
the complete GTU syllabus for the course on Mathematics-1, which is common to all
the engineering branches.
Objective
The crisp and complete explanation of topics will help students easily understand the
basic concepts. The tutorial approach (i.e., teach by example) followed in the text will
enable students develop a logical perspective to solving problems.
Features
Each topic has been explained from the examination point of view, wherein the theory
is presented in an easy-to-understand student-friendly style. Full coverage of concepts
is supported by numerous solved examples with varied complexity levels, which is
aligned to the latest GTU syllabus. Fundamental and sequential explanation of topics
are well aided by examples and exercises. The solutions of examples are set following a
‘tutorial’ approach, which will make it easy for students from any background to easily
grasp the concepts. Exercises with answers immediately follow the solved examples
enforcing a practice-based approach. We hope that the students will gain logical
understanding from solved problems and then reiterate it through solving similar
exercise problems themselves. The unique blend of theory and application caters to
the requirements of both the students and the faculty. Solutions of GTU examination
questions are incorporated within the text appropriately.
xii Preface
Highlights
∑ Crisp content strictly as per the latest GTU syllabus of Mathematics-1
(Regulation 2018)
∑ Comprehensive coverage with lucid presentation style
∑ Each section concludes with an exercise to test understanding of topics
∑ Solutions of GTU examination questions included appropriately within the
chapters
∑ Rich exam-oriented pedagogy:
 Solved examples within chapters: 850+
 Unsolved exercises: 500+
 MCQs at the end of chapters: 300+
Chapter Organization
The content spans the following 10 chapters which wholly and sequentially cover each
module of the syllabus.
 Chapter 1 introduces Indeterminate Forms.
 Chapter 2 discusses Improper Integrals.
 Chapter 3 presents Gamma and Beta Functions.
 Chapter 4 covers Applications of Definite Integrals.
 Chapter 5 deals with Sequences and Series.
 Chapter 6 presents Taylor’s and Maclaurin’s Series.
 Chapter 7 discusses Fourier Series.
 Chapter 8 presents Partial Derivatives.
 Chapter 9 covers Multiple Integrals.
 Chapter 10 deals with Matrices.
Acknowledgements
We are grateful to the following reviewers who reviewed sample chapters of the book
and generously shared their valuable comments:
Prof. Bhavini Pandya SVIT, Vasad
Prof. Som Sahani Babaria Institute of Technology, Baroda
We would also like to thank all the staff at McGraw Hill Education (India), especially
Vibha Mahajan, Shalini Jha, Hemant K Jha, Tushar Mishra, Satinder Singh Baveja,
Taranpreet Kaur and Anuj Shriwastava for coordinating with us during the editorial,
copyediting, and production stages of this book.
Preface xiii
Our acknowledgements would be incomplete without a mention of the contribution of

all our family members. We extend a heartfelt thanks to them for always motivating
and supporting us throughout the project.
Constructive suggestions for the improvement of the book will always be welcome.
Ravish R Singh
Mukul Bhatt
Publisher’s Note
Remember to write to us. We look forward to receiving your feedback,
comments and ideas to enhance the quality of this book. You can reach us at
info.india@mheducation.com. Please mention the title and authors’ name as the
subject. In case you spot piracy of this book, please do let us know.
Roadmap to the Syllabus
Mathematics-1
Subject Code: 3110014
Unit-I
Indeterminate Forms and L’Hôspital’s Rule.
Improper Integrals, Convergence and divergence of the integrals, Beta and Gamma
functions and their properties.
Applications of definite integral, Volume using cross-sections, Length of plane
curves, Areas of Surfaces of Revolution.
CHAPTER 1: Introduction to Some Special Functions
Go To
CHAPTER 2: Improper Integrals
CHAPTER 3: Gamma and Beta Functions
CHAPTER 4: Applications of Definite Integrals
Unit-2
Convergence and divergence of sequences, The Sandwich Theorem for Sequences,
The Continuous Function Theorem for Sequences, Bounded Monotonic Sequences,
Convergence and divergence of an infinite series, geometric series, telescoping
series, nth term test for divergent series, Combining series, Harmonic Series,
Integral test, The p-series, The Comparison test, The Limit Comparison test, Ratio
test, Raabe’s Test, Root test, Alternating series test, Absolute and Conditional
convergence, Power series, Radius of convergence of a power series, Taylor and
Maclaurin series.
CHAPTER 5: Sequences and Series

Go To
CHAPTER 6: Taylor’s and Maclaurin’s Series
xvi Roadmap to the Syllabus
Unit-3
Fourier Series of 2l periodic functions, Dirichlet’s conditions for representation
by a Fourier series, Orthogonality of the trigonometric system, Fourier Series
of a function of period 2l, Fourier Series of even and odd functions, Half range
expansions.
Go To CHAPTER 7: Fourier Series
Unit-4
Functions of several variables, Limits and continuity, Test for non existence of a
limit, Partial differentiation, Mixed derivative theorem, differentiability, Chain rule,
Implicit differentiation, Gradient, Directional derivative, tangent plane and normal
line, total differentiation, Local extreme values, Method of Lagrange Multipliers.
Go To CHAPTER 8: Partial Derivatives
Unit-5
Multiple integral, Double integral over Rectangles and general regions, double
integrals as volumes, Change of order of integration, double integration in polar
coordinates, Area by double integration, Triple integrals in rectangular, cylindrical
and spherical coordinates, Jacobian, multiple integral by substitution.
Go To CHAPTER 9: Multiple Integrals
Unit-5
Elementary row operations in Matrix, Row echelon and Reduced row echelon
forms, Rank by echelon forms, Inverse by Gauss-Jordan method, Solution of
system of linear equations by Gauss elimination and Gauss-Jordan methods. Eigen
values and eigen vectors, Cayley-Hamilton theorem, Diagonalization of a matrix.
Go To CHAPTER 10: Matrices

UNIT-1
Chapter 1. Indeterminate Forms

Chapter 2. Improper Integrals
Chapter 3. Gamma and Beta Functions
Chapter 4. Applications of Definite Integrals
CHAPTER
1
Indeterminate
Forms
chapter outline
1.1 Introduction
1.2 L’Hospital’s Rule
1.3 Type 1 : 0 Form
0
1.4 •
Type 2 : Form
•
1.5 Type 3 : 0 × • Form
1.6 Type 4 : • − • Form
1.7 Type 5 : 1•, • 0, 0 0 Forms
1.1 IntroductIon
We have studied certain rules to evaluate the limits. But some limits cannot be evalu-
ated by using these rules. These limits are known as indeterminate forms. There are
seven types of indeterminate forms:
0 ∞
(i) (ii) ∞ (iii) 0 × ∞ (iv) ∞ − ∞
0
(v) 1∞ (vi) 0° (vii) ∞°
These limits can be evaluated by using L’Hospital’s rule.
1.2 L’HospItaL’s ruLe
Statement If f (x) and g (x) are two functions of x which can be expanded by Taylor’s
series in the neighbourhood of x = a and if lim f ( x) = f (a ) = 0, lim g ( x) = g (a ) = 0,
x→a x→a
then
1.2 Chapter 1 Indeterminate Forms
f ( x) f ′ ( x)
lim = lim
x→a g ( x) x → a g ′ ( x)
Proof Let x=a+h
f ( x) f ( a + h)
lim = lim
x → a g ( x) h → 0 g ( a + h)
h2
f (a ) + hf ′ (a ) + f ″ (a ) + .......
= lim 2! [By Taylor’s theorem]
h→ 0 h2
g (a ) + hg ′ (a ) + g ″ (a ) + .......
2!
h2
hf ′ (a ) + f ″ (a ) + .......
= lim 2! [Q f (a ) = 0, g (a ) = 0]
2
h→ 0 h
hg ′ (a ) + g ″ (a ) + .......
2!
h
f ′ (a ) + f ″ (a ) + .......
= lim 2!
h→ 0 h
g ′ (a ) + g ″ (a ) + .......
2!
f ′(a)
=
g ′(a)
f ′ ( x)
= lim , provided g ′ (a ) ≠ 0.
x → a g ′ ( x)
Note
The following standard limits can be used to solve the problems:
sin x tan x ax −1
(i) lim =1 (ii) lim =1 (iii) lim = log a
x→0 x x→0 x x→0 x
1
 1
x
ex − 1
(iv) lim =1 (v) lim(1 + x) x = e (vi) lim 1 +  = e
x→0 x x→0 x →∞  x
sin −1 x sin h x tan −1 x

(vii) lim =1 (vii) lim =1 (ix) lim =1
x→0 x→0 x x→0 x
x
0
1.3 type 1 : Form
0
Problems under this type are solved by using L’Hospital’s rule considering the fact
that
0
0
f ( x) f ′ ( x)
lim = lim if lim f ( x) = 0 and lim g ( x) = 0.
x→ a g ( x) x → a g ′ ( x) x→ a x→ a
example 1
(1 + x) n − 1
Prove that lim = n.
x→0 x
Solution
(1 + x) n − 1 0 
Let l = lim  0 form 
x→0 x
n(1 + x) n −1
= lim [Applying L’Hospital’s rule]
x→0 1
=n
example 2
xe x - log (1 + x )
Evaluate lim .
x Æ0 x2
Solution
xe x − log (1 + x) 0 
x→0 x2
1
e x + xe x −
1+ x 0 
= lim  0 form 
x→0 2x
[Applying L’Hospital’s rule]
1
e x + e x + xe x +
(1 + x) 2
x→0 2
3
=
2
example 3
x log x − ( x − 1)
Evaluate lim .
x →1 ( x − 1) log x
Solution
x log x - ( x - 1) È0 ˘
Let l = lim Í 0 form ˙
x Æ1 ( x - 1) log x Î ˚
1
x◊
+ log x - 1
= lim x [ Applying L’Hospital’s rule]
x Æ1 1
log x + ( x - 1) ◊
x
log x È0 ˘
= lim Í 0 form ˙
x Æ1 1 Î ˚
log x + 1 -
x
1
x Æ1 1 1
+
x x2
1
=
2
example 4
e x - e - x - 2 log(1 + x )
Evaluate lim .
x Æ0 x sin x
Solution
e x - e - x - 2 log(1 + x ) È0 ˘
Let l = lim Í 0 form ˙
x Æ0 x sin x Î ˚
1
e x + e- x - 2 ◊
1+ x È0 ˘
= lim Í 0 form ˙ [ Applying L’Hospital’s rule]
x Æ0 sin x + x cos x Î ˚
2
e x - e- x +
(1 + x )2
= lim [ Applying L’Hospital’s rule]
x Æ 0 cos x + cos x - x sin x
2
=
2
=1
0
0
example 5
e x + e- x - x2 - 2
Evaluate lim . [Winter 2016; Summer 2014]
x Æ0 sin 2 x - x 2
Solution
e x + e- x - x2 - 2 È0 ˘
Let l = lim 2 2 Í 0 form ˙
x Æ0 sin x - x Î ˚
e - e- x - 2 x
x
x Æ 0 2 sin x cos x - 2 x
e x + e- x - 2
x Æ0 2( - sin x )sin x + 2 cos2 x - 2
e x + e- x - 2
= lim
x Æ0 -2 sin 2 x + 2 cos2 x - 2
e x + e- x - 2
= lim
x Æ 0 2 cos 2 x - 2
e x - e- x
x Æ 0 ( -2)2 sin 2 x
e x - e- x
= lim
x Æ 0 - 4 sin 2 x
e x + e- x
= lim [Applying L’Hospital’ss rule]
x Æ 0 -8 cos 2 x
2
=
-8
1
=-
4
example 6
2x - 1
Evaluate lim 1
.
x Æ0
(1 + x) 2 -1
Solution
2x - 1 È0 ˘
Let l = lim 1 Í 0 form ˙
x Æ0 Î ˚
(1 + x) 2 -1
2 x log 2
= lim 1
[ Applying L’Hospital’s rule]
x Æ0
1 -
(1 + x ) 2
2
= 2 log 2
example 7
Evaluate lim
(
log 1 + kx 2 ).
x→0 1 − cos x
Solution
l = lim
(
log 1 + kx 2 ) 0 
Let
x→0 1 − cos x  0 form 
1
⋅ 2kx
= lim 1 + kx 2 [Applying L’Hospital’s rule]
x→0 sin x
1
= 2k lim
( )
sin x
x→0
1 + kx 2 ⋅
x
 sin x 
= 2k Q lim = 1
x→0 x 
example 8
log (1 + x 3 )
Evaluate lim .
x Æ0 sin3 x
Solution
log (1 + x 3 ) È0 ˘
x Æ0 sin x Î ˚
1
3
⋅ 3x 2 [Applying L’Hospital’s rule]
= lim 1 + x
x → 0 3 sin 2 x cos x
2
Ê x ˆ 1
= lim Á ˜
x Æ 0 Ë sin x ¯ (1 + x 3 ) cos x
È x ˘
=1 Í xlim = 1˙
Î Æ0 sin x ˚
0
0
example 9
x − xx
Evaluate lim
x →1 1 + log x − x
.
Solution
x − xx 0 
x →1 1 + log x − x
x − e x log x
= lim
x →1 1 + log x − x
1 − e x log x (1 + log x)
= lim [Applying
g L’Hospital’s rule]
x →1 1
−1
x
 1
−e x log x (1 + log x) 2 − e x log x  
 x
x →1 1
− 2
x
=2
example 10
xy - yx
Evaluate lim .
xÆ y xx - yy
Solution
xy − yx 0 
x→ y x x − y y
yx y −1 − y x log y
x→ y x x (1 + log x) − 0
y y − y y log y
=
y y (1 + log y )
1 − log y
=
1 + log y
example 11
sin ( x cos x )
Evaluate lim .
xÆ
p cos( x sin x )
2
Solution
sin ( x cos x) 0 
x→
p cos ( x sin x)
2
cos ( x cos x)(cos x − x sin x)

= limp [Applying L’Hospital’s rule]
x→ − sin ( x sin x)(sin x + x cos x)
2
p
=
2
example 12
cos 2 π x
Evaluate lim 2 x . [Winter 2015]
x→
1 e − 2 xe
2
Solution
cos 2 p x 0 
Let l = lim1  0 form 
x→
2
e 2 x − 2 xe
2 cos p x ( −p sin p x)
x→
1 2e 2 x − 2 e
2
−p sin 2p x 0 
= lim1  0 form 
x→ 2( e 2 x − e )
2
−2p 2 cos 2p x
x→
1 2 ⋅ 2e 2 x
2
p2
=
2e
example 13
1 − cos (θ − α ) 1 2
Prove that θlim = sec α .
→α (sin θ − sin α ) 2 2
Solution
1 - cos(q - a ) È0 ˘
q Æa (sin q - sin a ) Î ˚
0
0
sin (q − a ) [Applying L’Hospital’s rule]

= lim
q →a 2 (sin q − sin a ) cos q
sin (q − a ) 0 
= lim  0 form 
q →a (sin 2q − 2 sin a cos q )
cos (q − a ) [Applying L’Hospital’s rule]
= lim
q →a 2 cos 2q + 2 sin a sin q
cos 0
=
2 cos 2a + 2 sin a sin a
1
=
2(1 − 2 sin 2 a ) + 2 sin 2 a
1
=
2 − 2 sin 2 a
1
=
2 cos 2 a
1
= sec 2 a
2
example 14
5 sin x − 7 sin 2 x + 3 sin 3 x
Evaluate lim .
x→0 tan x − x
Solution
5 sin x − 7 sin 2 x + 3 sin 3 x 0 
x→0 tan x − x
5 cos x − 14 cos 2 x + 9 cos 3 x
x→0 sec 2 x − 1
−5 sin x + 28 sin 2 x − 27 sin 3 x 0 
= lim  0 form  [Applying L’Hospital’s rule]
x→0 2 sec 2 x tan x
sin x sin 2 x sin 3 x
−5 + 56 ⋅ − 81
= lim x 2 x 3x
x→0 tan x
2 sec 2 x ⋅
x
−5 + 56 − 81  sin nx tan x 
= Q lim = 1 and lim = 1
2 x→0 nx x → 0 x 
= −15
example 15
3
2
2 x 2 − 2e x + 2 cos ( x 2 ) + sin 3 x
Evaluate lim .
x→0 x2
Solution
3
2
2 x 2 − 2e x + 2 cos x 2 + sin 3 x 0 
x→0 x2
2
3
3 1
4 x − 2e x (2 x) − 2 sin x 2  x 2  + 3 sin 2 x cos x
2  0 
= lim  0 form 
x→0 2x
2 2  3
3 1 1 3

4 − 4 (e x + xe x ⋅ 2 x) − 3  x cos x 2 ⋅ x 2 + sin x 2  + 6 sin x cos 2 x − 3 sin 3 x
 2 2 x 
= lim
x→0 2
3
2
sin x x
4 − 4 − lim ⋅
=
x→0 2 x x
2
3
1 1 sin x 2  3

= − ⋅ lim ⋅x 2
2 2 x→0 3
2
Qlim sin x = 1
x  x→0 3 
=0  x2 
example 16
A  sin kx k 
Evaluate lim
x→0 x2  sin lx − l  .
Solution
A  sin kx k 
Let l = lim −
x→0 x 2  sin lx l 
l sin kx − k sin lx 0 
= A lim  0 form 
x→0 lx 2 sin lx
A l sin kx − k sin lx
= lim
l x→0 sin lx
x2 ⋅ lx
lx
0
1.3 Type 1: Form 1.11
0
A l sin kx − k sin lx 0   sin lx 

= lim
2 x→0  0 form  Q lim = 1
l x3 x→0 lx 
A lk cos kx − kl cos lx 0 
= lim  0 form  [Appling L’Hospital’s rule]
l 2 x→0 3x 2
A −lk 2 sin kx + kl 2 sin lx
l 2 x→0 6x
A  sin kx sin lx 
= 2 lim  −lk 2 ⋅ ⋅ k + kl 2 ⋅ ⋅l
6l x → 0  kx lx 
 sin lx 
=
A
6l 2
(
−lk 3 + kl 3 ) Q lim
x→0
sin kx
kx
= 1 = lim
x → 0 lx 
=
Ak 2
6l
(
l − k2 )
example 17
1
2
x tan x
Evaluate lim 3
.
x→0
x 2
(e − 1)
Solution
x tan x È0 ˘
x Æ0 Î ˚
(e x - 1) 2
x x tan x
= lim 3
⋅
x→0 x
(e x − 1) 2
x x  tan x 
= lim ⋅ lim  
x→0 3 x→0  x 
(e x − 1) 2
3
 x 2  tan x 
= lim  x  Q lim = 1
x → 0  e − 1 x→0 x 
x 1
Now, lim x
= lim x [Applying L’Hospital’s rule]
x→0 e −1 x → 0 e
=1
3
3
 x 2
\ lim  x  = (1) 2 = 1
x → 0  e − 1
Hence, l=1
example 18
x
log sec x cos
Evaluate lim 2.
x →0 log
x cos x
sec
2
Solution
x
log sec x cos
Let l = lim 2
x → 0 log cos x
x
sec
2
x x
log cos log sec
= lim 2 ⋅ 2
x → 0 log sec x log cos x
x  − log cos x 
log cos  
= lim 2 ⋅ 2
x → 0 ( − log cos x ) log cos x
2
 x
log cos

= lim  2 0 
x→0 log cos x   0 form 
 
1  1 x
⋅ − sin 
x x  2 2 
log cos cos
lim 2 = lim 2
Now, x → 0 log cos x x→0 1 [Applying L’Hospital’s rule]
( − sin x)
cos x
x
tan
= lim 2
x → 0 2 tan x
 x
tan
1 2  x 
= lim  ⋅
x→0 4 x   tan x 
 
2 
1
=
4
0
1.3 Type 1: Form 1.13
0
2
 x
log cos 2
 2   1 1  tan x 
∴ lim  =   = Q lim = 1
x→0 log cos x  4 16 x→0 x 
 
1
Hence, l=
16
example 19
1
(1 + x) x − e e
Prove that lim =− .
x→0 x 2
Solution
1
(1 + x) x − e 0 
x→0 x
1
log(1+ x )
ex −e
x→0 x
1
log(1+ x )  1 1 
ex  − x 2 log (1 + x) + x (1 + x) 
  0 
= lim  0 form 
x→0 1
= lim (1 + x) x lim
1
[ − log (1 + x) ⋅ (1 + x) + x ] 0 
x→0 x→0 x (1 + x)
2  0 form 
 − log (1 + x) − 1 + 1 0  [Applying L’Hospital’s rule]

= e lim    0 form 
x→0
 2 x + 3x 2
 1 
  − 1 + x  
= e lim  
x→0
 2 + 6x 
 
e
=−
2
example 20
2n
Prove that lim

( 1− x −1 ) = 2− n .
n
x→0
(1 − cos x )
Solution
2n 2n
Let l = lim
( 1− x −1) ⋅( 1− x +1 )
n 2n
x→0
(1 − cos x ) ( 1 − x + 1)
(1 − x − 1) 2 n
= lim
 x
( )
x→0 n
2n
 2 sin
2
 1− x +1
2
( − x) 2 n 2n
= lim 2n
⋅
 x
( )
x→0 2n 2n
2n  sin  1− x +1
 2
2n
 x 
 
= lim  2 
2n
{
Q ( − x) 2 n = ( − x) 2 } = x2n 
n
( )  
x→0 2n
x 1− x +1
 sin 
2
1
=
2n
example 21
x 2 + 2 cos x − 2
Evaluate lim .
x→0 x sin x
Solution
x 2 + 2 cos x − 2 0 
x→0 x sin x
2 x − 2 sin x 0 
= lim  0 form  [Applying L’Hospital’s rule]
x→0 sin x + x cos x
2 − 2 cos x
x → 0 cos x + cos x − x sin x
=0
0
1.3 Type 1: Form 1.15
0
example 22
e x sin x − x − x 2
Evaluate lim .
x→0 x 2 + x log(1 − x)
Solution
e x sin x − x − x 2 0 
x → 0 x 2 + x log(1 − x )
e x sin x + e x cos x − 1 − 2 x 0 
= lim  0 form  [Applyying L’Hospital’s rule]
x→0 x
2 x + log(1 − x) −
1− x
e (sin x + cos x) + e x (cos x − sin x) − 2
x
x→0 1 1 x
2− − −
1 − x 1 − x (1 − x) 2
2(e x cos x − 1) 0 
= lim  0 form 
x→0  1  x
2 1 − −
 1 − x  (1 − x) 2
2(e x cos x − e x sin x)
x→0  1  1 2x
2 − 2− −
 (1 − x)  (1 − x) (1 − x)3
2
2
=−
3
exercIse 1.1
x 2 log a − a 2 log x 1
1. Prove that lim 2 2
= log a − .
x →a x −a 2
log(1 − x 2 )
2. Prove that lim = 2.
x → 0 log cos x
1 − tan x
3. Prove that limπ = 2.
x→
4
1 − 2 sin x
e x − 1 + 2x
x → 0 log(1 + x 2 )
a + x tan−1 a 2 − x 2
5. Evaluate lim .
x →a
a−x
 tan−1 a 2 − x 2 
Hint : lim(x + a) as x → a, a − x → 0 
 x →a
a2 − x 2 
[ans.: 2a]
1− x
e x + loge
6. Evaluate lim e .
x →0 tan x − x  1
 Ans.: − 2 
 
1 − x + log x
7. Evaluate lim .
x →1
1 − 2x − x 2
[ans.: –1]
x
xe − log(1 + x) 3
8. Prove that lim = .
x →0 x2 2
e x − 1 + 2x
x → 0 log(1 + x 2 )
1 − tan x
x→
π
4
1 − 2 sin x
3x − 12 − x 8
x →3
2 x − 3 19 − 5x 69
alog x − x a
12. Prove that lim = log .
x →1 log x e
x − a + x −a 1
2 2
x →a
x −a 2a
tan x − x 1
x →0 x3 3
1.4 type 2: Form
Problems under this type are also solved by using L’Hospital’s rule considering the
fact that
f ( x) f ′ ( x)
lim = lim if lim f ( x) = ∞ and lim g ( x) = ∞.
x →a g ( x) x →a g ′ ( x) x →a x →a
•
1.4 Type 2: Form 1.17
•
example 1
log x
Prove that lim = 0, (n > 0).
x →∞ x n
Solution
log x ∞ 
Let l = lim  ∞ form 
x →∞ xn
1
= lim xn −1 [Applying L’Hospital’s rule]
x →∞ nx
1 1
= lim n
n x →∞ x
=0
example 2
log sin x
Evaluate lim .
x →0 cot x
Solution
log sin x ∞ 
x→0 cot x
1
⋅ cos x
= lim sin x [Applying L’Hospital’s rule]
x → 0 − cosec 2 x
= lim − ( cos x sin x )

x→0
=0
example 3
 π
log  x − 
 2
Prove that lim = 0.
π tan x
x→
2
Solution
Ê pˆ
log Á x - ˜
Ë 2¯ È• ˘
Let l = lim Í • form ˙
xÆ
p tan x Î ˚
2
Ê 1 ˆ
Á p˜
ÁË x - ˜¯
2
xÆ
p sec 2 x
2
cos2 x È0 ˘
= lim Í 0 form ˙
xÆ
p
x-
p Î ˚
2
2
2 cos x ( - sin x )
xÆ
p 1
2
=0
example 4
log ( x – a )
Prove that lim = 1.
x → a log ( a x – a a )
Solution
log ( x - a ) È• ˘
Let l = lim Í • form ˙
x a
xÆa log (a - a ) Î ˚
1
( x - a)
xÆa 1 . x
a log a
a x - aa
 x 
. lim a − a 0 
a
1
= lim x  0 form 
x → a a log a x→a  x − a 
 
x
1 . lim a log a
= [Applying L’Hospital’s rule to second term]
a a log a x → a 1
1
= a ⋅ a a log a
a log a
=1
example 5
Prove that lim log x tan x = 1.
x→0
Solution
Let l = lim log x tan x

x→0
log tan x ∞ 
= lim  ∞ form 
[Change of base property]
x→0 log x
•
1.4 Type 2: Form 1.19
•
1 . 2
sec x
= lim tan x [Applying L’Hospital’s rule]
x→0 1
x
x .
limsec 2 x  tan x 
= lim
Q lim = 1
x → 0 tan x x → 0 x→0 x 
=1
example 6
Evaluate lim log tan x tan 2 x .
x →0
Solution
Let l = lim log tan x tan 2 x

x→0
log tan 2 x ∞ 
= lim  ∞ form 
x→0
log tan x
1
⋅ 2 sec 2 2 x
= lim tan 2 x [Applying L’Hospital’s rule]
x→0 1
⋅ sec 2 x
tan x
tan x
⋅ sec 2 2 x
= lim x
x → 0 tan 2 x
⋅ sec 2 x
2x
 tan x 
=1 Q lim = 1
x→0 x 
example 7
sin h−1 x
Prove that lim = 1.
x →∞ cos h −1 x
Solution
sin h −1 x
Let l = lim
x →∞ cos h −1 x
= lim
(
log x + x 2 + 1 ) ∞ 
 ∞ form 
x →∞
log ( x + x 2
− 1)
1  1 
⋅ 1 + ⋅ 2 x
= lim
( x + x +1  2 x +1
2 2
) 
x →∞
1  1 
⋅ 1 + ⋅ 2 x
(x + x2 − 1 )  2 x2 − 1 
x2 + 1 + x
= lim
(x + x2 + 1 ) x2 + 1
x →∞
x2 − 1 + x
(x + x2 − 1 ) x2 − 1
x2 − 1
= lim
x →∞
x2 + 1
1
1−
x2
= lim
x →∞
1
1+ 2
x
=1
example 8
1 2 3 x
e x + e x + e x + ... + e x
Prove that lim = e − 1.
x →∞ x
Solution
1 2 3 x
e x + e x + e x + ... + e x
Let l = lim
x →∞ x
1 
 1 
x
e x 1 −  e x  
    1
= lim 
[Sum of GP]
1
⋅
x →∞ x
1− e x
1
e x (e − 1) 1
= lim 1 ⋅
x →∞ x
e −1
x
Putting 1
= y, when x → ∞, y → 0
x
(e − 1)e y y 0 
l = lim  0 form 
y→0 ey −1
•
1.4 Type 2: Form 1.21
•
(e − 1) ( ye y + e y )
y →0 ey
= e −1
example 9
xn
Prove that lim = 0.
x →∞ e kx
Solution
xn ∞ 
x →∞ e kx
nx n −1 ∞ 
= lim  ∞ form  [Applying L’Hospital’s rule]
x →∞ ke kx
n (n − 1) x n − 2 ∞ 
= lim  ∞ form  [Applying L’Hospital’s rule]
x →∞ k 2 e kx
Applying L’Hospital’s rule (n – 2) times in the above expression,
n (n − 1) (n − 2)...2 ⋅1
l = lim
x →∞ k n e kx
n!
= lim n kx
x →∞ k e
=0 [Q lim e kx = ∞]
x →∞
example 10
12 + 22 + 32 + ... + x 2 1
Prove that lim = .
x →∞ x3 3
Solution
12 + 22 + 32 + ... + x 2
Let l = lim
x →∞ x3
x ( x + 1) (2 x + 1)  n (n + 1) (2n + 1) 
= lim Q∑ n =
2

x →∞ 6 x3 6
2 x3 + 3x 2 + x
= lim
x →∞ 6 x3
3 1
2+ + 2
= lim x x
x →∞ 6
2
=
6
1
=
3
example 11
1
ex
Prove that lim = e2 .
x →∞ x
 1  x 
1 +  
 x 
Solution
ex È• ˘
Let l = lim x Í • form ˙
x Æ•
ÈÊ 1ˆ ˘
x Î ˚
ÍÁ 1 + ˜ ˙
ÍÎË x¯ ˙
˚
ex
= lim
x Æ• x2
Ê 1ˆ
ÁË 1 + x ˜¯
Taking logarithm on both sides,
È x2 ˘
Ê 1ˆ
log l = lim Í log e x - log Á 1 + ˜ ˙
x Æ• Í Ë x¯ ˙
Î ˚
È Ê 1ˆ˘
= lim Í x - x 2 log Á 1 + ˜ ˙
x Æ• Î Ë x¯˚
È1 Ê 1ˆ˘
= lim x 2 Í - log Á 1 + ˜ ˙
x Æ• Î x Ë x¯˚
1 Ê 1ˆ
- log Á 1 + ˜
x Ë x¯ È0 ˘
= lim Í 0 form ˙
x Æ• 1 Î ˚
x2
1 1 Ê 1ˆ
- 2 - -
1 ÁË x 2 ˜¯
x 1+
x Æ• 2
- 3
x
1.5 Type 3 : 0 × ∞ Form 1.23
1
1−
1
1+
= lim x
x →∞ 2
x
1 1
= lim
2 x →∞ 1
1+
x
1
=
2
1
Hence, l = e2
exercIse 1.2
log x
cot x
x →0
log(1 − x)
x →1 cot π x
logsin x cos x
x →0 x
log x cos
sin
2
2
4. Prove that limlogsin x sin 2 x = 1.
x →0
log(1 + e 3 x )
x →∞ x
log x 2
6. Prove that lim = 0. [Hint: Put x 2 = y ]
x → 0 cot x 2
xm
7. Prove that lim x = 0 (m > 0) .
x →∞ e
2 3 n
 1  1  1  1
  +   +   + LL +  
8. Prove that lim e e e e = 0.
n →∞ n
1
9. Prove that limlog x sin 2 x = .
x →0 2
1.5 type 3 : 0 × ∞ Form

To solve the problems of the type
lim [ f ( x) ⋅ g ( x)], when lim f ( x) = 0, lim g ( x) = ∞ (i.e. 0 × ∞ form)
x→ a x→ a x→ a
f ( x) g ( x)
We write lim [ f ( x) ⋅ g ( x) ] = lim or lim ⋅
x→a x→a 1 x→a 1
g ( x) f ( x)
0 ∞
These new forms are of the type or respectively, which can be solved using
0 ∞
L’Hospital’s rule.
example 1
Prove that lim x log x = 0.
x→0
Solution
Let l = lim x log x [0 × ∞ form]
x→0
log x ∞ 
= lim  ∞ form 
x→0 1
x
1
l = lim x [Applying L’Hospital’s rule]
x→0 1
− 2
x
= lim( − x)
x→0
=0
example 2
Prove that lim sin x log x = 0 .
x→0
Solution
Let l = lim sin x log x [0 × ∞ form]
x→0
log x ∞ 
= lim  ∞ form 
x→0 cosec x
1
= lim x [Applying L’Hospital’s rule]
x → 0 − cosec x cot x
tan x
= − lim sin x ⋅
x→0 x
tan x
= − lim sin x ⋅ lim
x→0 x→0 x  tan x 
Qlim = 1
=0
x→0 x 
1.5 Type 3 : 0 × ∞ Form 1.25
example 3
 a
Prove that lim 2 x ⋅ sin  x  = a.
x →∞ 2 
Solution
 a
Let l = lim 2 x ⋅ sin  x 
x →∞ 2 
1 1
Putting 2x = , t = x ,
t 2
When x → ∞, 2 x → ∞, t→0
sin at
l = lim
t →0 t
a sin at
= lim
t →0 at
 sin x 
= a ⋅1 Q lim = 1
x→0 x 
=a
example 4
 x
Evaluate lim log  2 −  cot( x − a ).
x→ a  a
Solution
 x
Let l = lim log  2 −  cot( x − a ) [0 × ∞ form]
x→a  a
Ê xˆ
log Á 2 - ˜
Ë a¯ È0 ˘
= lim Í 0 form ˙
x Æ a tan( x - a ) Î ˚
1 Ê 1ˆ
-
Ê x ˆ ÁË a ˜¯
ÁË 2 -
a ˜¯
xÆa sec 2 ( x - a )
1
=-
a
example 5
 1x 
Prove that lim  a − 1 x = log a . [Winter 2013]
x →∞  
Solution
Let l = lim(a − 1) ⋅ x
x →∞
1
x [0 × ∞ form]
= lim
(a − 1)
1
x 0 
x →∞ 1  0 form 
x
1
Putting = t , when x → ∞, t → 0
x
at − 1 0 
l = lim  0 form 
t →0 t
a t log a [Applying L’Hospital’s rule]
= lim
t →0 1
= log a
example 6
Ê p xˆ
Prove that lim tan 2 Á ˜ (1 + sec p x ) = - 2.
x Æ1 Ë 2 ¯
Solution
Ê pxˆ
Let l = lim tan 2 Á ˜ ( 1 + sec p x ) [• ¥ 0 form ]
x Æ1 Ë 2 ¯
1 + sec p x È0 ˘
= lim Í 0 form ˙
x Æ1 Ê pxˆ Î ˚
cot 2 Á ˜
Ë 2 ¯
p sec p x tan p x
x Æ1 Ê pxˆ Ê pxˆ p
2 cot Á ˜ Á - cosec 2
Ë 2 ¯Ë 2 ˜¯ 2
Ê ˆÊ ˆ
Á sec p x ˜ Á tan p x ˜
= - Á lim ˜ Á lim
x Æ1 2 px x Æ1 px ˜
Á cosec ˜Á cot ˜
Ë 2 ¯Ë 2 ¯
Ê ˆ
Á sec p ˜ tan p x È0 ˘
= -Á ˜ lim Í 0 form ˙
p
Á cosec 2 ˜
x Æ1
cot
px Î ˚
Ë 2 ¯ 2
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The strict attention to duty which is enforced in the Guards is
shown in the personal neatness of the men and the fine condition of
their trenches. Be the conditions never so bad, and water never so
scarce, a Guardsman in the field will always contrive to present a
clean appearance. If his uniform is stained and patched, his puttees
will be neatly tied and his boots, cumbrous though they are, will be
scraped clean of mud. The army knows that a well-turned-out
battalion, clean in appearance and punctilious about saluting, is a
good fighting battalion. It has been the ambition of the Guards to set
an example to the army in this respect.
The Guards trenches are famous all along our line. Deep, and well
made, and clean, and as safe as they can be made, they are named
after London streets familiar to the Guards on their “walkings out” on
Sunday evenings—Piccadilly and Bond Street and Edgware Road
and Praed Street. When the Guards are in their trenches they are
kept scrupulously neat and tidy; the greatest attention is paid to
hygiene, with the result that the plague of vermin and flies is
probably less felt here than in any other part of the line.
There is a fine disdain for the shirker in the Guards. The Guards
esteem it an honour to be able to fight with the Guards against the
Germans, and therefore among the men there is nothing but
contemptuous pity for the young fellow who prefers to loaf at home
to earning the proud right to say in after years: “I was with the
Guards in Flanders!” But for the Guards officer who should stay
away from his battalion in the front line for any reason whatsoever,
even to do useful duties in the rear, the officers have nothing but the
most withering disdain.
It is no use arguing the question. It is no use pointing out to them
that a man who is by temperament not a fighter can render better
service to the country by serving on a Staff or doing other work
behind the fighting-line. The only reply is that “they cannot imagine
how the fellow can stay away from his battalion at a time like this.”
You feel that they are often unjust in such wholesale condemnation,
but you cannot help admiring the real Guards spirit which is reflected
in this attitude of mind.
Their spirit is one of the most jealous exclusiveness. It is apparent
in their mental attitude as well as in their dress. The Guards officers
have succeeded in investing even their prosaic service khaki with
one or two little touches that render their uniform quite distinct from
that of officers of the line. In the first place, the different Guards
regiments retain their distinctive button groupings in the service
tunics of the officers, the buttons being arranged in ones for the
Grenadier, in twos for the Coldstream, in threes for the Scots
Guards, and in fours for the Irish Guards. It is etiquette that the
buttons should be of dulled bronze, and as small and as unobtrusive
as possible. No badges are worn on the collar, and the badge on the
cap is silver and diminutive in size. Many Guards officers affect
excessively baggy breeches, cut like full golfing-knickers, and worn
with puttees. They are certainly distinctive, but they can hardly be
said to be becoming, and are liable to get sodden and heavy from
the wet in the trenches, I am told.
The Guards in the field judge life by two standards—the Guards’
standard and other men’s standard. There is nothing offensive to the
rest of the army in their carefully studied exclusiveness. They are
genuinely and generously appreciative of the undying gallantry
displayed by line regiments. They show themselves friendly and
companionable neighbours in the trenches, and stout and reliable
comrades in action. But you will find that what they are seriously
willing to concede to other regiments they will never allow to the
Guards. They have no criticism to offer if a line battalion surrenders
after a most gallant stand against overpowering odds, but if you
probe their minds you will find that they would naturally expect a
Guards battalion in similar circumstances to fight to the last man.
And the remarkable thing is that, if you examine their records in this
war, you will find that this is the standard the Guards have set and
lived up to.
Indeed, the Guards’ spirit is not of this war. It is of another age. It
is as old as chivalry itself. It was to the British Guards, if you
remember, that the Frenchmen at Fontenoy said: “Messieurs les
Anglais, tirez les premiers!” and as far as our Guards are concerned,
a similar incident might occur in this war. You will never hear a
Guardsman disparage the German as a fighter. He thinks the
German is a bad sportsman, and, remembering Belgium and the
Lusitania, he has a fierce joy in fighting him. But he knows he is a
brave man, for our Guards, remember, saw the Prussian Guard
advancing in parade order to their death at Ypres in the face of a
perfect tornado of shot and shell.
“They were fine, big men all,” a Guards officer who witnessed that
last desperate attempt to break our line said to me, “and they walked
past the corpses of their dead comrades choking their line of
advance, and straight into our machine-gun fire like brave men that
were not afraid to die.”
It was the spirit of the Guards in Marlborough’s day that sent the
Guardsman William Lettler across the river at Lille to cut the chains
of the drawbridge. It was the same spirit that carried the Guards
forward at Talavera with such impetuosity that a catastrophe was
only narrowly averted, that at Waterloo welded them into a solid wall
of steel. It was the Guards’ spirit that transformed the little Irishman,
Michael O’Leary, into an epic hero; that inspired the Coldstreams at
Ypres, the Scots Guards at Festubert, to fight to the last man.
This book is not a history, and I must leave the story of the
Guards’ achievements in this war to an abler pen than mine. From
the outset they have been in the very thick of the fighting. At Mons
we find the famous Guards Brigade—2nd Grenadiers, 2nd and 3rd
Coldstream and 1st Irish Guards—with the Second Division, and
with the First Division the 1st Coldstream and the 2nd Scots Guards.
With Sir Henry Rawlinson’s Seventh Division in Belgium—that
splendidly gallant division of whose exploits in the early days of the
war we heard so little—were the 1st Grenadiers and the 2nd Scots
Guards. At Mons the Guards battalions played their part gallantly in
beating back the desperate attempt of the Germans to overwhelm
“French’s contemptible little army” with vastly superior numbers.
No battle honour will figure more gloriously in years to come on
the colours of the four battalions of the Guards Brigade than
Landrecies. The magnificent stand which the brigade, under General
Scott-Kerr, made in this little town averted a disaster and inflicted
enormous losses on the enemy. The Germans delivered a surprise
attack on the place in the mist and darkness of the night of August
25, hoping, by dint of tremendously superior numbers, to overwhelm
the Guards and burst through our line. The Guards hastily
improvised a defence, and throughout the night, through hours of
bloody and desperate hand-to-hand fighting in the narrow streets,
held their own. At last the Germans realized that the surprise had
failed, and withdrew, leaving their dead piled up in ramparts on the
cobblestones.
With the rest of the British Army, sorely pressed and exhausted,
but not beaten, the Guards fell back from Mons. We hear of the
Guards Brigade again in the woods of Villers-Cotterets, fighting a
desperate rear-guard action, engaged at close quarters, as they love
to be. Here the Irish Guards, on active service for the first time in this
war—and right gallantly have they acquitted themselves—lost their
Colonel, Lieutenant-Colonel Morris, a man as brave as he was big.
When fortune changed, and to the stern ordeal of the retreat and
the bad news from Belgium succeeded the spirited pursuit of Von
Kluck falling back baffled from Paris, once more we find the Guards
in the centre of things. They fought in the battle of the Marne, and
advanced with the rest of the army to the Aisne. Carrying out Sir
John French’s historic order to “make good the Aisne,” the Guards
Brigade had a stiff fight at Chavonne, but managed to cross the river
at this place, after overcoming severe German resistance in the
woods, by ferrying a battalion over the stream.
In the first great struggle about Ypres in October, 1914, the Guards
—like every other British regiment engaged there, be it said—gave
of their best in the defence of our line. The 2nd Scots Guards,
holding the trenches at Kruseik, north-east of Zandvoorde, came in
for the brunt of the smashing attempt of the Germans to pierce the
line of the Seventh Division. The enemy actually managed to break
through, but the gap was closed and the bulk of the storming party
killed or made prisoner. The 2nd Scots Guards counter-attacked with
splendid dash, and the German attempt failed, but in a subsequent
vigorous assault by the enemy the gallant battalion was all but
exterminated. That was on October 25. On October 31—by Sir John
French’s own admission, the critical day of the battle—when the
Germans broke through the line of the First Division, the 1st
Coldstreams held on till the end, and were practically destroyed.
Meanwhile, the Guards Brigade, which had come into line on the
previous evening, was fighting desperately on the left of the First
Division. I have already mentioned in my chapter on Sir John French
how the 2nd Worcester Regiment, by its gallant charge at Gheluvelt,
saved the British Army on that fateful 31st of October. For a time the
peril was averted. But after a short respite from their persevering
efforts to obey the Emperor’s command to win Ypres at all costs, the
Germans attacked again on November 6, this time against the Klein
Zillebeke position, defended by the 2nd and the Guards Brigades
and a French division—the Ninth—under General Moussy.
General Scott-Kerr, who commanded the Guards Brigade at Mons,
wounded at Villers-Cotterets, relinquished the command to
Brigadier-General the Earl of Cavan, who came out from England to
take up the appointment. Lord Cavan commanded at Ypres. The
achievements of the Guards Brigade in this war will for ever be
associated with his name. A short, stoutly built little man, there is
nothing particularly suggestive of the great soldier in his personal
appearance, but a few minutes’ talk with him will show you the fine
courage in his keen eyes, the tremendous virility in his language and
gestures, that bespeak the leader of men.
Cavan is the Guards’ spirit incarnate. All his admiration goes to the
fearless man. I wish I could tell you in his own words, as he told me,
the now familiar story of how Mike O’Leary of the Irish Guards won
the V.C. at Cuinchy. The General (who believes that where the
Guards are there he should also be) witnessed the incident himself.
He tells how he saw O’Leary, right ahead of his company, dash up
one bank and kill the Germans there, then dash up another and kill
the Germans there, then, going round the back, seize two Germans
working a machine-gun by the scruff of the neck, and with either
hand gripping their collars firmly, call to his comrades to relieve him
of his prisoners. “A most extraordinary fellow,” says the General. “By
rights he should have been killed a dozen times.”
Lord Cavan’s own fearlessness and complete indifference to
danger are a by-word in the army. His officers swear by him. His men
adore him, regarding him, with true Guards’ exclusiveness, as a
treasured possession, a peculiar acquisition of the Guards. More
than once he has been mentioned in despatches. This is what the
Commander-in-Chief, on the recommendation of Sir Douglas Haig,
wrote of his conduct at the first battle of Ypres: “He was conspicuous
for the skill, coolness, and courage with which he led his troops, and
for the successful manner in which he dealt with many critical
situations.” Lord Cavan has no enemies, I believe, and no one who
has seen him in the field will think that what I have written in praise
of him is excessive.
A sudden German attack on November 6 drove back the troops on
the left of the Guards Brigade, which was left exposed. A splendid
charge by the Household Cavalry brought a British cavalry brigade to
fill the gap on the left of the Guards, and the next day the Guards
delivered a successful counter-attack, but could not retain the
ground they had won against the overwhelming German odds.
On November 11 the final desperate effort of the Germans to
break through to the sea, in the shape of the attack of the Prussian
Guard, failed, and the battle came to a close. The First Brigade, with
which were the 1st Coldstream and the 1st Scots Guards, with the
rest of the First Division, stemmed the tide and threw the flower of
the German Army back in confusion. The First Brigade left its
commander on that blood-stained field in the person of the gallant
Brigadier-General Fitzclarence, V.C., who, as I have told elsewhere,
was the author of the famous order to the 2nd Worcesters that saved
the day at Ypres on October 31.
The next serious fighting in which the Guards were involved was
to the south, in the wet and dreary black country opposite La
Bassée. In December the First Division was ordered up to Givenchy
to relieve the Indian Corps, which had been having a very bad time,
and on December 21 the First Brigade found itself holding the
trenches from Givenchy down to the La Bassée Canal. The prompt
intervention of this fine division enabled our line, from which the
Indians had been partially forced back, to be re-established.
This ugly and sinister region from Givenchy to Cuinchy, situated
on the other side of the La Bassée Canal, was destined to be the
home of the Guards for many months, and the scene of some of
their most heroic exploits in this war. On January 24 the First
Brigade, under Brigadier-General Cecil Lowther, found itself holding
the line in the Cuinchy brick-fields opposite the famous La Bassée
Railway triangle formed by the Béthune-La Bassée line and the
Lens-La Bassée line, which joins the first in two branches. The 1st
Scots Guards and the 1st Coldstreams were in the trenches. On the
25th the Germans opened a heavy bombardment of the Guards’
trenches, which were practically destroyed. The line was broken, and
the Germans managed to secure a footing in the brick-fields. A
counter-attack, delivered with great gallantry by the Black Watch, the
Cameron Highlanders, and the King’s Royal Rifles, succeeded in
partially clearing our second line. The First Brigade lost heavily, and
was relieved during the night.
On February 1 Cavan’s Guards Brigade was holding the line
through the Cuinchy brick-fields. At half-past two in the morning the
2nd Coldstreams were driven out of their trenches, but managed to
hold out till daylight in a position close to their old trench. A counter-
attack launched in the small hours by some of the Irish Guards and
Coldstreams was checked by the enemy’s rifle-fire.
Another counter-attack was arranged for 10.15 a.m. It was
preceded by a splendid artillery preparation—the kind of ruthless and
accurate rain of high-explosive projectiles that puts a fine heart into
men waiting to attack. Then the storming-party went forward, a grand
array of big, stalwart men in whose great hands their rifles with
bayonets fixed seemed as light and as inconsiderable as toothpicks.
Captain A. Leigh Bennett led the way at the head of fifty of the 2nd
Coldstreams, bent on “getting their own back”; following came
Second-Lieutenant F. F. Graham with thirty Irish Guards, with whom
went one Michael O’Leary in the front line, and a party of Royal
Engineers with barbed wire in rolls, and sandbags, to “organize” the
trenches that might fall into our hands.
It was a magnificent piece of work. The Guards were irresistible.
They swept like an avalanche over the lost trench, bayoneting their
way. All the ground lost was retaken, and another trench besides,
while two machine-guns and thirty-two prisoners fell into our hands.
It was here that Michael O’Leary performed his prodigious exploit.
But the achievements of the Guards were not confined to brilliance
in the open with the bayonet. They proved themselves well-
disciplined, uncomplaining, resourceful, and patient through the long
winter months in the trenches in this sordid region, which vies with
the Ypres salient as being the ugliest, wettest, and most depressing
portion of our whole line.
Neuve Chapelle saw the 1st Grenadiers and 2nd Scots Guards in
line with the Seventh Division. The latter stages of that historic fight
made great demands on the courage and tenacity of the troops
engaged, and these two famous battalions maintained their high
reputation for both. The Guards were not engaged in the second
battle of Ypres. Their services were required farther south, where the
attack on the Fromelles ridge, to support the French “push” in the
Artois, was preparing. In the operations which began on May 9, and,
continuing with intervals until the middle of June, resulted in the gain
of a mile or two of front and the capture of several hundred
prisoners, all the brigades in which the Guards are serving were
concerned.
After the attack by the Seventh Division on May 15 on the German
trenches south of Richebourg l’Avoué, the greater part of a company
of the 2nd Scots Guards, including Captain Sir Frederick
Fitzwygram, was found to be missing. Presently word came down to
the brigade—I think from the Canadians, who had taken over the line
here—that some Scots Guards’ graves had been located. Would the
brigade send up an officer to investigate?
An officer was despatched. He was destined to elucidate the
mystery of the missing company. He did not find Fitzwygram, who
had been wounded and captured. But he found the dead bodies of
sixty Scots Guards lying huddled together in the open, the centre of
a grim circle of some 200 German corpses, and close by two rough
white crosses marking the spot where the Canadians had laid two
Scots Guards officers to rest.
The Scots Guards, who had advanced side by side with the
Border Regiment, had outdistanced their fellows. They were found
dead, amid heaps of empty cartridge cases, with their rifles still
grasped in their stiffened fingers, in the place where they had last
been seen through a drifting haze of high-explosive vapours,
standing shoulder to shoulder together under a murderous fire
poured in on them from three sides. Soaked by the rain and
blackened by the sun, their bodies were not beautiful to look upon,
but monarch never had nobler lying-in-state than those sixty
Guardsmen dead on the coarse grass of the dreary Flanders plain.
It has been my privilege to have seen a good deal of the Guards in
this war. You would scarcely recognize in these battle-stained
warriors the spruce Guardsmen of St. James’s and the Park. The
first Guardsmen I met in this war were a battalion of Irish Guards and
a battalion of the Coldstreams on an evening in May, as they were
marching down a road near Chocques towards the firing-line. Their
creased caps and stained khaki, their dull green web equipment and
short brown rifles, made them look at the first superficial glance like
any other Regular troops. But something about their stride, the way
they bore themselves, their alignment (though they were marching
easy), made me look again. That magnificent physique, that brave
poise of the head, that clear, cool look of the eye—that could only be
of the Guards!
They had come a long way. It was a close, warm evening, and the
roads were a smother of choking dust. The Guards wore their caps
pushed back off their foreheads, and their tunics unbuttoned at the
neck, showing a patch of white skin where the deep tan of their faces
and necks ended. The perspiration poured off them in streams. It
traced little channels in the dust that lay thick on their sunburnt
cheeks. Every now and then a man, with a grunt, would wipe the
sweat from his eyes, and in the same motion administer that little
hoist to his pack that is peculiar to the British soldier marching with a
full load.
Many a time, on German manœuvres, I have passed a regiment
on the march, like these Guardsmen, in the stifling dust of a summer
day. I have been all but choked by the sour odour which the breeze
has wafted over from the marching men, and have been only too
glad to follow the advice of the old hands to put the wind between
them and me. But these British Guardsmen, grimy and travel-stained
though they were on the outside, were clean of body, and the air
about them was pure. Looking at them closer, I saw that under the
dust their haversacks were neatly packed and fastened, their uniform
well-fitting and whole, their puttees beautifully tied. These things may
seem trifles to you who will read this in the sheltered atmosphere of
England, where one man in khaki with a gun seems as another. But
they are the mark of a good battalion, and, noting them on that dusty
French road, with the guns drumming faintly in the distance and an
aeroplane droning aloft, I knew that the trenches for which those
troops were bound would be well held.
One of the most stirring military spectacles it has ever been my
good-fortune to witness was a parade of some battalions of the
Guards before the Commander-in-Chief, Lord Kitchener, and M.
Millerand, the French Minister of War. I have seen many reviews of
the Prussian Guard before the German Emperor, both in Berlin and
in Potsdam. My eye has been fascinated by the perfect precision of
the movements of the Prussian drill, the long lines of heads thrown
stiffly over at the left shoulder at the same angle, of feet flung
forward in the Paradeschritt with such mathematical exactitude that
they seemed as one movement, of white-gloved hands swinging to
and fro in absolute accord.
I have watched the rippling line of the French infantry swing past
the saluting-point at Longchamps reviews with a wiry elasticity that
gave better promise of efficiency in the field than the stiff precision of
the Prussian, and have delighted in the brilliant array of colours, the
red and blue and gold and silver against the deep green background
of the historic racecourse.
But I have never seen, and never wish to see, a more inspiring
picture than those four battalions of Guards drawn up in their drab
khaki on a heath in Flanders over against the Tricolour and the
Union Jack flying side by side. An ancient military tradition, a high
purpose and perfect physical condition, never combined to produce
a more sublime spectacle of troops than this. There was no display,
no searching after cheap effects. The Guards were there in their
khaki, as they had come from the trenches; the officers carried no
swords; the colours were guarded in churches at home. The
Grenadiers and Coldstreams had their drums and fifes; the Scots
Guards their pipers, wearing the proud red tartan of the ancient
House of Stuart. The four battalions stood there in four solid
phalanxes, unbeautiful and undecorative, save when, to the crash of
the opening bars of the “Marseillaise,” three distinct ripples ran
through those serried ranks, and with a dazzling flash of steel the
Guards presented arms.
Memories of Mons and Landrecies, of Klein Zillebeke and Cuinchy
and Festubert, went shuddering by, pale shadows escaping from the
prison of the imagination, as the stalwart giants of the King’s
Company of the Grenadiers led off the march past. The drums and
fifes crashed through “The British Grenadiers” again and again and
again before the serried files of men, marching with an iron tread that
fairly shook the earth, had all gone by, and the skirling of the pipes
proclaimed the approach of the Scots Guards.
There were faces in that procession like faces on a Greek frieze,
fighters all, radiant with youth and strength and determination to
conquer or readiness to die, men who had looked Death in the eyes,
and, in that they had withstood the ordeal, had risen above man’s
puny fears of the Unknown. It was a spectacle to thrill a soldier, to
inspire a poet, to make an Englishman vibrate with pride at the
thought that these are his brothers.
The army in the field loves the Guards. It is not jealous of their
exclusiveness, for the Guards have shown in this war that they are
not content to rest upon their laurels. The army trusts the Guards, for
it knows that, when in a critical hour Wellington’s voice shall be
heard once more above the storm crying, “Stand up, Guards!” the
Guards will rise again in a solid wall of steel, as invulnerable as their
phalanx at Waterloo.
CHAPTER XIII
THE ARBITERS OF VICTORY
“... My guns are better than the German guns ... for
instance, my 15-inch shell is equivalent to their 17-inch. The
issue is now one between Krupp’s and Birmingham.”
(Field-Marshal Sir John French to Mr. James O’Grady, M.P.,
quoted in the Daily News, August 23, 1915.)
“Too-too! Too-too! Too-too!”
“‘Ul-loh?” (wearily).
“Too-too! Too-too! Too-too!” (with insistence).
“‘Ul-loh?” (with vexation). “‘Ul-loh? ‘Ul-loh?”
The sounds issued forth from a low, cramped dug-out, where a
perspiring orderly, squatting on a box, huddled over a crepitating
telephone-receiver—not the “gentlemanly article” of your City office
or my lady’s boudoir, but a Brobdingnagian kind of instrument.
Fragments of conversation drifted out of the hole:
“’Oo? ... I can’t ’ear yer.... Oh! Yessir! Yessir! Yessir!”
Then a sentence was bawled and repeated from mouth to mouth
till it reached the orderly standing at the end of the trench. “The
Major of the Blankshires sends ’is compliments to Captain X, and
there’s a German working-party be’ind the village clearly visible. Will
Captain X send a few rounds over?”
The Captain turned wearily to the subaltern by his side
(Cambridge O.T.C., out since March, keen as mustard). “Did you
ever see such fellows?” he said. Then, to the orderly: “My
compliments to the Major, and we have been watching that working-
party for the past half-hour. Unfortunately, it is out of range. But tell
him, you can, that we have just dispersed another working-party over
by the bridge!”
This message is shouted from mouth to mouth, the telephone
toots again, but even before the Major in his dug-out a mile away
has had his answer, the battery is called up once more from another
quarter, with the request to “turn on for a bit” in some other direction.
So it goes on all day, and every day. The guns are the big brothers
of the trenches. To them the front line, like the small boy in a London
street row, appeals when bullied by the German artillery. To them the
men in the trenches look for protection against working-parties
preparing new “frightfulness,” against spying aircraft, against undue
activity on the part of the minenwerfer.
The gunners keep guard over the front line in a paternal and
benevolent, not to say patronizing spirit. Their business it is to find
places from which they can keep an eye on the enemy, watch the
effect of their shells, and see what the enemy’s guns are doing. No
matter that these places are exposed; no matter that the Germans
search for them with their guns like caddies “beating” the heather for
a lost ball; no matter that, sooner or later, they will be brought down
about the observers’ ears. Observation is a vital part of artillery work.
It saves British lives; it kills Germans.
When German “frightfulness” oversteps the bounds of what is
average and bearable, “retaliation” is the word that goes back to the
guns. When there are bursts of German “liveliness” going on all
along the line, the battery telephones (so the men in the fire-trenches
tell me) are so busy that to call up a battery is like trying to get the
box-office of the Palace Theatre on the telephone at dinner-time on a
Saturday night.
This word “retaliation” has a fine ring about it. To men with nerves
jaded by a long spell of shelling with heavy artillery it means a fresh
lease of endurance. To the least imaginative it conjures up a picture
of the Germans, exulting in their superiority of artillery, watching in
fascination from their parapets their “Jack Johnsons” and “Black
Marias” ploughing, among eddies of black smoke, great rifts in our
trench-lines, starting back in terror as, with a whistling screech, the
shells begin to arrive from the opposite direction.
Nothing puts life into weary troops like the sound of their own
shells screaming through the air and mingling with the noise of the
enemy’s guns. Nothing in the same way puts a greater strain on
men, even the most seasoned and hardened troops, than to have to
sit still under a fierce bombardment, and to know that their guns
must remain inactive because ammunition is limited to so many
rounds a day per gun.
Big shell exploding on a road in France. German soldiers

in foreground.
Our success at Hooge on August 9 showed us, albeit on a very
small scale, what our infantry, adequately supported by artillery, can
achieve. The Hooge affair was of quite minor importance, and had
next to no bearing on the general situation. It was, however, of the
deepest interest to all of us on the Western front, for at this
engagement, almost for the first time, the guns had a free hand in
the matter of ammunition. It was due to their thorough and
devastating preliminary bombardment and their splendid support
afterwards that our infantry were able to recapture the lost trenches,
and to hold them against all German attempts to win them back.
The battalions who took part in the attack against Hooge were
brimful of gratitude towards the gunners. Not only had the guns done
the work of cutting the barbed wire and of destroying the enemy’s
machine-guns and defences in the most thorough manner, but the
timing of the different operations of the artillery—the preliminary
bombardment, the lifting on the front line and then the tir de barrage
(to prevent the Germans from sending up reinforcements)—was
absolutely perfect. As a result, the attacking files were able to push
on freely with no fear of coming under their own fire, for, with each
position won, the rain of shells lifted and poured down farther ahead.
A bombardment like this makes men optimistic. The infantry,
waiting in the paling darkness before dawn for the preliminary
bombardment to do its work, were elated as the heavy shells went
plumping into the German trenches, as the curtain of smoke from the
high-explosive drifted thicker and thicker in the twilight.
The men’s comments were fierce and heartfelt. “Ah! they’re getting
it at last, the ——s! That’ll learn ’em! Boom!” (as a shell exploded
with a deafening crash). And when the storming-party went forward it
was irresistible. The men “had their tails up,” as the army says. Not
only did they hurl the Germans out of their trenches, but they held
the positions. Every time the Germans massed for a counter-attack
our guns were ready for them, and swept the attack away before it
had properly developed.
I was able to appreciate the elation of the men who fought at
Hooge, because I had already seen, in past months, the evil effects
of the shortage of shells. As I talked to these gallant fellows, full of
their stories of the fight, I kept thinking of a position I had visited
months before where, day after day, the Germans kept up an almost
incessant bombardment with shells of all calibres. As so much of
their strategy, their action was aimed at achieving a moral effect, like
their senseless raids on undefended English seaside resorts. At this
particular position they had no chance of breaking through our line,
yet, so much do they believe in the effect of high-explosives in
undermining the moral of troops, that they fired away ammunition
from their guns in the most reckless fashion.
I spent an afternoon in a dug-out there with a Colonel, a fine,
hearty body of a man, while the shells passed almost continuously
over our heads or burst noisily, with whizzing fragments of hot metal,
in our immediate vicinity. He expressed himself very sagely on this
question of shells. “Of course,” he said, “perpetual bombardment like
this is a great strain on the men. But they are stout fellows, and they
stand it all right. What does upset them, though, is the silence of our
guns. My men see that a little energy on the part of our guns
promptly reduces the Germans to silence. If we give the Germans
shot for shot, and sometimes, for a change, two shells to their one,
they become as meek as lambs. What my men lack is the heartening
sound of our own shells. When our guns do retaliate, their fire is
feeble, and soon dies away. A well-fed man is a better fighter than a
fellow with an empty belly. So troops who know that they have the
weight of metal behind them may be counted upon to make a better
showing than men who feel that they are not properly supported.”
But I do not intend to go further into the shell controversy here.
Hooge showed us that the tide has turned. The shells which our
guns fired that day were better than anything the gunners had ever
seen before. The issue, in the Commander-in-Chief’s expressive
phrase, is now between Krupp’s and Birmingham. Birmingham will
win if England understands that, were every man left in England and
every woman from this moment to give their services to the making
of munitions of war, even then our Generals could not
conscientiously say they were certain that the supply would be
sufficient.
In our former wars the gunners reaped the richest harvest of
decorations, for they most often found themselves in a tight corner,
and the rule of the Royal Artillery is, “Never lose a gun!” If you want a
contrast between the war of movement and the war of positions,
between August, 1914, and August, 1915, you have but to look at
the gunners then, when the war was in the open, and now, when the
belligerents have sat down to a siege. The foaming horses, the
jingling accoutrements, the mud-bespattered guns, the gunners
falling man by man about their gun—this splendid picture of war,
painted on many a glorious page of our military history, has been
replaced—temporarily, I know—by as strange a blend of peace and
war as can be seen in the field to-day.
Not that in the main the gunner’s task to-day is less perilous than
before. Most of the shells that the Germans send over night and day
are hunting for our batteries, searching for them principally with the
aid of little maps provided by the German air observers. The
Germans have to reckon with a resourceful lot of men in our
gunners, however, and so skilfully are the guns hidden that more
often than not the enemy’s projectiles plump harmlessly into fields,
and cause no greater damage than the massacre of a few dozen
buttercups and daisies.
But the gunner is toujours en vedette. He must stay with his gun. If
his battery is firing, he can take no thought of safety. He must
expose himself. Shells or no shells, he must go about his duty. Thus,
though the day of the race forward with the guns is over for the
moment, the gunners’ casualties go on steadily, in little driblets, it is
true, but nevertheless mounting up to a long roll of honour that will
show future generations that, in the long months of trench warfare,
the Royal Artillery, true to its grand traditions, did its work as
steadfastly and gallantly as in the thrilling days of the retreat from
Mons and the advance on the Marne.
The gunners have had a difficult task in Flanders. Since the battle
of the Marne the great offensives on our front have been with the
enemy. The Germans, even when advancing, never forget for a
moment that he who goes forward may also have to fall back, and
their striving has always been to secure for themselves good
defensible positions from which their guns can dominate our lines.
The flatness of the country has made the work of artillery
observation both perilous and difficult, and the heroism with which
the observation officers exposed themselves in the fighting round
Ypres, at Neuve Chapelle, and at Festubert, to mention but a few
concrete cases, is a chapter by itself in the roll of fame of the Royal
Artillery in this war. The time has not yet come when one may write
freely of this most thrilling and perilous of duties at the front, the
observing for the guns, but I have seen the observation officers at
their work, and feel very strongly that one cannot pass over their
gallantry and tenacity in silence.
I have witnessed the growth of our artillery; with it I have passed
from the discouraging days of the shell shortage to the brighter era
that dawned at Hooge. I have spent many absorbingly interesting
days with the gunners in the batteries, with the observation officers
at their stations. I have seen in action every type of gun, every type
of howitzer, big and small, used in our army to-day. I know the voices
of them all—the short, sharp report of the field-gun, the ear-splitting
crash of the 4·7, the air-shaking roar of the heavy howitzer, whose
shell in its passage through the air over one’s head produces long-
drawn-out reverberating waves of sound that resemble nothing on
earth unless it be the echo of a fast motorcar rushing through miles
of empty, narrow streets.
But because I have thus seen the work of our guns in the field,
because I have been allowed to penetrate the arcana of gunnery,
and to enter holy places where few save those of the craft are
suffered to enter, I find myself in a dilemma. Despite the ceaseless
activity of German spies in peace and war, there are still many
blanks in the German information about our artillery. Far be it from
me to fill them in! Permit me, therefore, to leave the guns hidden
beneath their covers, cunningly stowed away from prying eyes in
their emplacements.
In a battery you get a good idea of the detachment of modern war.
The effect of the war of positions is to hold off from the guns the
rough-and-tumble of fighting. It happens occasionally, as in the case
of the London Territorial “Heavies,” surrounded and captured during
the second battle of Ypres, that the tide of battle sweeps up and
washes round the guns. But as a general rule the guns, in their
positions behind the front, are several miles from the actual firing-
line. Their rôle is practically that of the artillery of a besieging force,
and they thereby acquire a fixity of tenure which must be particularly
sympathetic to the gunners who remember the awful ordeal of the
guns on the retreat from Mons.
A skilful fencer who has a good reach and a long foil can keep his
opponent at a distance without moving his position, without losing his
calm. The gunners in this war of positions are using a foil several
miles long. Their blade is constantly in play, but the enemy is kept at
arm’s length. Therefore, the life of the gunners is probably more
ordered and stable than the life of any other formation at the front.
In a London newspaper office some years ago I used to know a
cable operator who on Saturday nights, when the wires were clear,
was wont to pass the time by playing cable chess with the operators
at Queenstown or New York. Many an evening I have sat by his side
and watched him, poring over his chessboard, with one hand on the
cable-key ready to transmit his move across the wires. The sense of
long distance, which that quiet cable-room, with its shaded lights, its
absorbed chess enthusiast, and its ticking instrument, used to
convey, has come back to me in these batteries at the front, where
death goes and comes over a stretch of miles.
Among my notes of a visit to a battery of field-guns, I find the
following observation: “The coolness, the savage Sachlichkeit (a
comprehensive German word which may be rendered by ‘business-
likeness’) of the gunners; exposed to the constant menace of death
at long range themselves, their one thought in life is to ‘get back’ at
the enemy on the distant horizon.”
This studied efficiency in the business of slaying is indeed the
dominant impression left on the mind by a view of one of these
batteries in action. The officers, always neat and trim, are nonchalant
and laconic as they snap out in monosyllables the orders that carry
with them winged death; the men serving the gun—just a handful of
dust-coloured figures about some ironwork—go about their work
swiftly and silently. A word with the telephone orderly as to range
and angle and shell, a brisk order, a yellow-painted shell is whipped
smartly out of its neat caisson standing open by the gun, its doors
flung wide like two arms, as though to say, “Help yourself!” there is a
snapping and a clicking of bolts and blocks, then the voice of the
sergeant: “All ready to fire, sir!”
The gun is laid on its quarry. No gun fires without an object.
Somewhere in the sun-bathed flats beyond, where the horizon is
smudged in blue behind a curtain of dancing heat, there are men
whose glass of life is all but run out. How often have I thought on this
picture, in the few seconds’ interval between that short phrase, “All
ready to fire, sir!” and the brisk command: “No. 1 gun, fire!” ... A
group of Germans, not the fierce helmeted Huns in exquisitely neat
uniforms that our artists love to show us in the illustrated papers, but
vague figures in ill-fitting dirty grey that makes them look like
scavengers, with trousers, showing a band of white lining, turned up
over their boots, curiously long tunics unbuttoned at the throat, and
little round caps, all stained with mud and sweat, their heads
beneath shaved until the skin shows greyish through the pale
stubble—a group of vague Unknowns waiting for death.
Perhaps they are walking up a communication trench, plodding in
Indian file, silent; maybe they are sitting outside a dug-out talking of
the things which form the staple conversation of the men on both
sides out here—their food, their duties, their spell of rest from the
trenches, their leave, their sergeant. All are marked down by the
Power that regulates these things—those that are to die, those that
shall survive. Even now one is taking that step forward that shall
save his life, another has tarried an instant, and for that instant’s
delay he shall die. The cigarette that the officer in our battery is
lighting shall give the doomed ones a few seconds’ respite.
The shell is in the gun, bearing in its shining and beautifully turned
case that which shall release a never-ending succession of events
springing from the widows and orphans which this shot shall make.
In a second now it will be sped, in another second its work will be
done. There will be telegrams—ill-omened and feared visitants in
these days of war—arriving in German homes, in the officer’s flat on
the third-floor of a dull street in a garrison town, in the tenement
dwelling in a Berlin slum, in a farm in Pomerania, in a wooden
cottage on a pine slope in the Black Forest.

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Mathematics-1, GTU–2018 Ravish R

Probability and Statistics (GTU) 2nd Edition Ravish R

Advanced Engineering Mathematics (Gujarat Technological

Advanced Engineering Mathematics (Gujarat Technological

Basic Electrical Engineering 3rd Edition Ravish R.

Calculus Third Edition Gujarat Technological University

Circuit Theory and Networks—Analysis and Synthesis (MU

Circuit Theory and Networks Analysis and Synthesis 2nd

Linear Algebra and Vector Calculus (2110015) Gujarat

Ravish R Singh is presently Director at Thakur Ramnarayan College

Mukul Bhatt is presently Assistant Professor at Department of

McGraw Hill Education (India) Private Limited

McGraw Hill Education Offices

Cover Designer: APS Compugraphics

Visit us at: www.mheducation.co.in

Soumya and Siddharth

Points to Remember 3.36

9.7 Area by Double Integrals 9.141

Our acknowledgements would be incomplete without a mention of the contribution of

CHAPTER 1: Introduction to Some Special Functions

CHAPTER 5: Sequences and Series

Go To CHAPTER 7: Fourier Series

Go To CHAPTER 8: Partial Derivatives

Go To CHAPTER 9: Multiple Integrals

Go To CHAPTER 10: Matrices

Chapter 1. Indeterminate Forms

1.2 L’HospItaL’s ruLe

sin −1 x sin h x tan −1 x

cos ( x cos x)(cos x − x sin x)

sin (q − a ) [Applying L’Hospital’s rule]

A l sin kx − k sin lx 0   sin lx 

 − log (1 + x) − 1 + 1 0  [Applying L’Hospital’s rule]

Prove that lim

1.4 type 2: Form

= lim − ( cos x sin x )

Let l = lim log x tan x

Let l = lim log tan x tan 2 x

1.5 type 3 : 0 × ∞ Form

Big shell exploding on a road in France. German soldiers

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