LIMITS, CONTINUITY AND DIFFERENTIABILITY

EXERCISE – I

SINGLE CHOICE CORRECT

2
e x  cos x
1. lim is equal to
x 0 x2
(a) 3/2 (b) 1/2
(c) 2/3 (d) none of these

 log(1  ax )  log(1  bx )
 , x0
2. If f ( x )   x and f(x) is continuous at x = 0, then the value of k is
 k x 0
(a) a  b (b) a  b
(c) log a  log b (d) none of these

1  cos 2 x
2
3. The value of lim is
x 0 x
(a) 1 (b) – 1
(c) 0 (d) none of these
x
4. The set of points where f ( x )  is differentiable, is
4 |x|
(a) ( , +) (b) (0, +)
(c) (, 0)  (0, +) (d) none of these

 | x 3|, x 1

5. The function f ( x )   x 2 3 x 13
 4  2  4 , x  1
which of the following is not true?
(a) is continuous at x = 1 (b) is continuous at x = 3
(c) is differentiable at x = 1 (d) f (3) exists
x4
 x  6
6. The value of lim   is
x   x  1 

(a) e5 (b) e 4
(c) e 2 (d) 0

7. The function f(x) = max (1  x ), (1  x ), 2 for all real x. Then f(x) is
(a) continuous at all points
(b) differentiable at all points
(c) differentiable at all points except at x = 1
(d) continuous at all points excepts at x = –1, at x = 1; where it is discontinuous
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

8. f(x) = | [x] x| in 1  x  2 is (where [.] denotes greatest integer function)

(a) continuous at x = 0 (b) discontinuous at x = 0
(c) differentiable at x = 0 (d) continuous at x = 2

x (1  a cos x )  b sin x
9. If lim  1 then
x 0 x
(a) a = b (b) a + b = 0
(c) 2a = b (d) none of these

c  dx
 1 
10. If a, b, c, d are positive, then lim  1   is equal to
x   a  bx 

(a) ed/b (b) ec/a

(c) e(c + d)/(a + b) (d) e

 tan 1 x ; if x  1

11. The domain of derivative of the function f x    1 is
 2  x  1 ; if | x |  1

(a) R  0 (b) R  1

(c) R   1 (d) R   1, 1

log x n  [ x ]
12. lim , n  N , (where [.] denotes greatest integer less than or equal to x)
x  [ x]
(a) has value –1 (b) has value 0
(c) has value 1 (d) does not exist

[a  n nx  tan x ] sin nx

13. If lim  0 , where n is non zero real number, then a is equal to
x 0 x2
n 1
(a) 0 (b)
n
1
(c) n (d) n 
n

14. Given that f 2   6 and f 1  4 , then lim

 
f 2h  2  h 2  f 2
h 0 
f hh 2
 1  f 1

3
(a) does not exist (b) is equal to 
2
3
(c) is equal to (d) is equal to 3
2
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

x x x  x 

15. The value of lim cos  cos  cos ...... cos n  is
n  2 4 8 2 
sin x x
(a) 1 (b) (c) (d) none of these
x sin x

 x 2  1 ; x  0, 2
sin x ; x  n , n  Z 
16. If f x    and g x    4 ; x  0 , then lim g f x  is
 2 ; otherwise  5 x 0

 ; x  2

(a) 5 (b) 6 (c) 7 (d) 1

 sin[ x ]
 [ x ]  1 ; for x  0

 cos  [ x ]
 2
17. If f x    ; for x  0 ; where [.] denotes the greatest integer function, then in
 [ x]
 k ; at x  0



order that f be continuous at x = 0, the value of k is
(a) equal to 0 (b) equal to 1
(c) equal to –1 (d) indeterminate

1x
 f 1  x 
18. Let f : R  R be such that f 1  3 and f 1  6 , then lim   equals
x 0
 f 1 

(a) 1 (b) e 1 2 (c) e2 (d) e

  x 
1  tan  [1  sin x ]
  2 
19. lim is

x   x  3
2 1  tan  [   2 x ]
  2 

1 1
(a) (b) 0 (c) (d) 
8 32
20. Let f ( x )  lim sin 2n x , then number of points where f(x) is discontinuous, is/are
n 

(a) 0 (b) 1
(c) 2 (d) infinitely many
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

 1 2 n 
21. lim    ...   is equal to
n   1  n 2 1 n 2
1 n 2

(a) 0 (b) –1/2 (c) 1/2 (d) 1

22. lim (6n + 5n)1/n is equal to

n  

(a) 6 (b) 5 (c) e (d) does not exist

x
23. Let f x  be defined for all x  0 and be continuous. Let f x  satisfy f    f x   f y  for
y 
all x, y and f e  = 1, then

(a) f x   ln x (b) f x  is bounded

 1
(c) f    0 as x  0 (d) xf x   1 as x  0
x

e tan x  e x
24. lim is equal to
x 0 tan x  x

(a) 1 (b) e
(c) e – 1 (d) 0

25. The value of lim tan 2 x  2 sin 2 x  3 sin x  4  sin 2 x  6 sin x  2  is equal to
x  / 2  
1 1
(a) (b)
10 11
1 1
(c) (d)
12 8
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

EXERCISE – II

IIT-JEE – SINGLE CHOICE CORRECT

 ax 2  b x  1
1. If the derivative of the function f ( x )   2 is continuous everywhere, then
bx  ax  4 x  1
(a) a  2, b  3 (b) a  3, b  2
(c) a  2, b  3 (d) a  3, b  2

x tan 2 x  2 x tan x
2. lim is
x 0 (1  cos 2 x ) 2
1 1
(a) 2 (b) –2 (c) (d) –
2 2

 1 3
| x | cos  15 x , x 0
3. Let f ( x )   x , then f(x) is continuous at x = 0 if k is equal to
 k , x 0
(a) 15 (b) 15 (c) 0 (d) 6

0 , x is irrational
4. The function f ( x )   is
1 , x is rational
(a) continuous at x = 1 (b) discontinuous only at 0
(c) discontinuous only at 0 and 1 (d) discontinuous every where

5. The number of points at which the function f(x) = | x  0.5 | + | x  1 | + tan x, does not
have a derivative in the interval (0, 2) is
(a) 1 (b) 2 (c) 3 (d) 4

6.  
The function f(x) = x 2  1 | x 2  3 x  2 |  cos | x | is not differentiable at
(a) 2 (b) 0 (c) 1 (d) 1

7. Let f : R  R be a function. Define g : R  R by g x   f x  for all x. Then g is

(a) onto if f is onto (b) one-one if f is one-one

(c) continuous if f is continuous (d) differentiable if f is differentiable

8. Let f ( x )  ( x /( x  3)) 3x . Then

(a) lim f ( x )  e 3 (b) lim f ( x )  1/ 64

x  x 1

(c) lim f ( x )  64 / 625 (d) lim f ( x )  3 / 4

x 2 x 2
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

x
9. For the function f ( x )  , x  0, f (0)  0 the left hand derivative f ' (0 ) and right hand
1  e1 / x
derivative f ' (0  ) are given by (respectively)
(a) 0, –1 (b) 1, 0 (c) 0, 0 (d) 1, –1

10. The function f ( x ) || x | 1 |, x  R is differentiable at all x  R except at the points

(a) 1, 0, –1 (b) 1 (c) 1, –1 (d) –1

 e 1 x   e 1 x 

11. If f x   x e 1 x   e  1 x  ; x  0 , then which of the following is true
 0 ; x 0
(a) f is continuous and differentiable at every point
(b) f is continuous at every point but is not differentiable everywhere
(c) f is differentiable at every point
(d) f is differentiable only at the origin

1/ x 2
 e x  e x  2 
12. lim  2

 is equal to
x 0
 x 
1/ 2
(a) e (b) e 1 / 4 (c) e 1/ 8 (d) e 1/ 12

13. Let g(x) be the inverse of an invertible function f(x) which is differentiable at x = c, then g´(f(x))
equals
1
(a) f ' (c ) (b) (c) f (c ) (d) none of these
f ' (c )

14. The function f ( x )  [ x ] 2  [ x 2 ] (where [y] is the greatest integer less than or equal to y) is
discontinuous at
(a) all integers (b) all integers except 0 and 1
(c) all integers except 0 (d) all integers except 1
15. If f(x) = [x sin x], where [x] represents the greatest integer  x; then f(x) is
(a) discontinuous at x = 0 (b) continuous in (– 1, 0)
(c) not differentiable in (–1, 1) (d) none of these
sin 4  [ x ]
16. Let f ( x )  , where [x] is the greatest integer  x, then
1 [x ]2
(a) f(x) is not differentiable at some points
(b) f (x) exists but is different from 0
(c) f (x) = 0 for all x
(d) f (x) = 0 but f is not a constant function

 ( 2 x  1)  
17. The function f(x) = [x] cos   (where [x] denotes the greatest integer function) is
 2 
discontinuous
(a) at all x (b) at all integer points
(c) at no x (d) at x which is not an integer
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

18. Which of the following functions is/ are differentiable at x = 0?

(a) cos( | x | )  | x | (b) sin( | x | )  | x |
(c) sin( | x | )  | x | (d) cos( | x | )  | x |

19. The left hand derivative of f(x) = [x] sin (x) at x = k (k an integer) is ([.] denotes greatest
integer function)
(a) (1)k (k  1)  (b) (1)k1 (k  1) 
(c) (1)k k (d) (1)k1 k

(cos x  1) (cos x  e x )
20. The integer n for which the lim is a finite and non-zero is
x 0 xn
(a) 1 (b) 2 (c) 3 (d) 4

ONE OR MORE THAN ONE CHOICE CORRECT

1  cos 5 x
1. The points of discontinuity of the function f x   are
1  cos 4 x
 
(a) x = 0 (b) x (c) x (d) x
2 4

x nx
m  m
2. lim n C x   1   equals to
n  n  n
mx mx m x 1
(a) . e m (b) . em (c) e0 (d)
x! x! me m x !

1
3. Given the function f x   , the points of discontinuity of the composite function
1  x 
y  f 3n x  , where f n x   fof ......of (n times) are
(a) 0 (b) 1 (c) 3n (d) 2

 1  1 
 x sin sin  ; x  0
4. Let f x    x  x sin 1 x  , then f(x) is
 0 ; x0

(a) both continuous and differentiable at x = 0
(b) continuous at x = 0
(c) neither continuous nor differentiable at x = 0
(d) f (0  0) fails to exist
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

a sin x  bx  cx 2  x 3
5. If L = lim exists and is finite, then
x 0 2 x 2 log1  x   2 x 3  x 4

3 3
(a) a  6, L   (b) a  6, L 
40 40
3
(c) b = 6, c = 0 (d) a  6, b  6, L 
40
6. Points of discontinuities of the function f(x) = 4x + 7 [x] + 2 log (1 + x), where [.] denotes the
integral part of x, is
(a) 0 (b) 1 (c) 3/2 (d) none of these

7. In the interval 0  x  2 the function f x   sin 2 x is not differentiable at

  3
(a) (b) (c)  (d)
4 3 2

8. If f x   x  1  [ x ] , where [x] = the greatest integer less than or equal to x, then

(a) f 1  0   1, f 1  0   0 (b) f 1  0   0  f 1  0 
(c) lim f x  exists (d) lim f x  does not exist
x 1 x 1

9. Let [x] denote the greatest integer less than or equal to x. Now g x  is defined as below:
     
[f x ] , x   0,    ,  
 2 2 ,
 
2 sin x  sin n x  sin x  sin n x
g x    where f x   ,
 3 , x
 2sin x  sin x  
n
sin x  sin n x
 2
n  R . Then

(a) g x  is continuous at x  , when n  1
2

(b) g x  is differentiable at x  , when n  1
2

(c) g x  is continuous but not differentiable at x , when n  1
2

(d) g x  is continuous but not differentiable at x  , when 0  n  1
2

10. Let f x   [tan 2 x ], where [.] denotes the greatest integer function. Then
(a) lim f x  exists (b) f x  is continuous at x = 0
x 0

(c) f 0   1 (d) f x  is not differentiable at x = 0

 LIMITS, CONTINUITY AND DIFFERENTIABILITY

EXERCISE – III

MATCH THE FOLLOWING

Note: Each statement in column – I has one or more than one match in column - II
1.

Column I Column II

I. f x   [sin x ]  [cos x ], x  [0, 2] , where [.] is greatest

integer function. Number of points where f x  is not A. 9
differentiable is equal to
II. Let f x  is a differentiable function such that
f x  y   f x . f y  for all x, y where f 0   0 . If f 0   3 ,
  1  B. 7
then f    is equal to (where [.] denotes greatest
  3 
integer function)

III. Let f x   [m  5 sin x ], x  0,  , m  I and [.] represents

greatest integer function. Then number of points at which C. 8
f x  is non differentiable is

IV. If lim
a  77 x  tan x  sin 7 x  0 , then [a] is equal to
D. 5
2
x 0 x
(where [.] represents greatest integer function) E. 3

REASONING TYPE

Direction: Read the following questions and choose:

(A) If both the statements are true and statement-2 is the correct explanation of
statement-1.
(B) If both the statements are true but statement-2 is not the correct explanation of
statement-1.
(C) If statement-1 is True and statement-2 is False.
(D) If statement-1 is False and statement-2 is True.
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

 e1 x  e 1 x
 ; x  0 . Then f x  has a jump discontinuity at x = 0.
1. Statement-1 : Let f x    e1 x  e 1 x
 0 ; x 0

Statement-2 : Since lim f x   1 and lim f x   1 .

x 0 x 0

(a) A (b) B (c) C (d) D

2. Statement-1 : The function f x   [ x  1] is discontinuous for all integral values of x in its

domain except at x = –1 (where [x] is greatest integer  x).
Statement-2 : [g(x)] will be discontinuous for all ‘x’ given by g(x) = I, where I is set of all
integers.
(a) A (b) B (c) C (d) D

3. Statement-1 : f x   | x |2 3 | x |  2 is not differentiable at 5 points.

Statement-2 : If graph of f x  crosses the x-axis at ‘m’ distinct points, then g x   f x  is

always non differentiable at least at ‘2m’ distinct points.
(a) A (b) B (c) C (d) D

4. Statement-1 : If a and b are positive and [x] denotes greatest integer  x , then
x b b
lim    .
x 0  a  x  a

Statement-2 : lim
x  0 , where {x} denotes fractional part of x.
x  x
(a) A (b) B (c) C (d) D

5. Statement-1 : If f x   x for all x  R , then f x  is continuous at x = 0.

Statement-2 : If f x  is continuous, then f x  is also continuous.

(a) A (b) B (c) C (d) D
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

LINKED COMPREHENSION TYPE

The relation between differentiability and continuity is defined as follows. If left hand
derivative and right hand derivative of the function is same and it is finite then the function is
continuous as well as differentiable. But if left hand and right hand derivatives are different
but finite then the function is continuous but not differentiable. If at least one left hand and
right hand derivative is infinite then the function may be continuous but not differentiable.
Now answer the following questions.
x
1. If f x    [t ] dt , then
0
(a) f x  is continuous as well as differentiable for x  N
(b) f x  is continuous but not differentiable for x  N
(c) f x  is not continuous and differentiable for x  N
(d) f x  is continuous and differentiable for x  Q
x
2. f x   is
1 2 1 x
(a) continuous (b) differentiable
(c) limit does not exist at x  N (d) continuous as well as differentiable

   3[ x ]  x 
cos 1  sgn   , x  0
3. f x     2 x  [  x ]  is

 0 , x0

(a) continuous at x = 0
(b) continuous and not differentiable at x = 0
(c) left and right hand limit does not exist
(d) none of these
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

EXERCISE – IV

SUBJECTIVE PROBLEMS

1
x 4 sin  x2
1. Evaluate lim x
x   1 | x |3

2. Is there any value c that will make the following functions :

(i) Continuous at x = 0 (ii) Differentiable at x = 0.
 sin 2 3 x  x  c x0
 , x0
(a) f ( x )   x 2 (b) f ( x )  
 c, x 0 cos x x0


sin 1 (1  { x }) . cos 1 (1  { x })
3. Let f ( x )  , then find lim f ( x ) and lim f ( x ) , where { } denotes
2{ x } . (1  { x }) x 0  x 0 

fractional part function.

  
sec 2  
     2  bx 
4. Evaluate lim sin 2   .
x 0
  2  ax  

5. If f(x) f(y) = f(x) + f(y) + f(xy) – 2; f(2) = 5 find f(3); being given that f(x) is a polynomial
function.

6. Find the limit

e x  e  x  2x
lim .
x 0 x  sin x

  x 2  1
  2 
 ; 0x 2
  x  1

1
7. Discuss the differentiability of f ( x )   ( x 3  x 2 ) ; 2x 3 .
4

 9  1  | 2  x |  ; 3  x  4
 4  | x  4 | 



8. If f(x) = | x |  1, then draw the graph of f(x) and fof(x) and also discuss their continuity and
2 3
differentiability. Also find derivative of fof  at x  .
2
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

9. Let f(x) = 1 + 4x  x2  x  R
max . {f (t ), x  t  x  1} ; 0  x  3

g( x)  
 min . ( x  3) ; 3 x5

Check the differentiability of g(x) for all x  [0, 5].

10. Determine the constants a, b and c for which the function

 1/ x
 (1  ax ) ; x0


f (x )   b ; x 0
 1/ 3
 (x  c) 1 ; x  0
 ( x  1)1/ 2  1

is continuous at x = 0.
 LIMITS, CONTINUITY AND DIFFERENTIABILITY

ANSWERS

EXERCISE – I

AIEEE-SINGLE CHOICE CORRECT

1. (a) 2. (b) 3. (d) 4. (a) 5. (d)

6. (a) 7. (a) 8. (a) 9. (b) 10. (a)

11. (d) 12. (a) 13. (d) 14. (d) 15. (b)

16. (d) 17. (a) 18. (c) 19. (c) 20. (d)

21. (b) 22. (b) 23. (a) 24. (a) 25. (c)

EXERCISE – II

IIT-JEE – SINGLE CHOICE CORRECT

1. (a) 2. (c) 3. (c) 4. (d) 5. (c)

6. (a) 7. (c) 8. (b) 9. (b) 10. (a)

11. (b) 12. (d) 13. (b) 14. (d) 15. (b)

16. (c) 17. (c) 18. (b) 19. (a) 20. (c)

ONE OR MORE THAN ONE CHOICE CORRECT

1.(a, b, c) 2. (a, d) 3. (a, b) 4. (b, d) 5. (b, c)

6. (a, b, c) 7. (c, d) 8. (a, d) 9. (a, b) 10. (a, b)

 LIMITS, CONTINUITY AND DIFFERENTIABILITY

EXERCISE – III

MATCH THE FOLLOWING

1. I-(D), II-(C), III-(A), IV-(B)

REASONING TYPE

1. (a) 2. (b) 3. (c) 4. (a) 5. (b)

LINKED COMPREHENSION TYPE

1. (b) 2. (a) 3. (a)

EXERCISE – IV

SUBJECTIVE PROBLEMS

1. 1

2. (a) For c = 9 continuous and differentiable

(b) For c = 1 continuous and non differentiable

2
/ b2
3. /2 and / 2 2 4. e a

5. 10 6. 2

7. Non-differentiable at x = 2 and x = 3.

3
8. fof(x) is continuous  x but not differentiable at x = {2, 1, 0, 1, 2}; at x  value of
2
derivative is –1

9. g(x) is non differentiable at x = 3

10. a = loge(2/3), b = 2/3, c = 1