BUSINESS

CALCULUS
MATH3042
CHAPTER 3
 Outline: Chapter 3
• 3-1 Introduction to Limits
Introduction to Limits.
 Definitions
 Theorem on
existence of a Limit
 Limit Properties
 Indeterminate Forms
  Difference Quotient
 3.1 Introduction to Limits
Functions and Graphs: Brief Review
• The graph of the function y = f(x) = x + 2 is the graph of the set
of all ordered pairs (x, f(x)).
• For example, if x = 2, then f(2) = 4 and (2, f(2)) = (2, 4) is a point
on the graph of f.
• Figure 1 shows and (-1, f(-1)), (1, f(1)), (2, f(2)) plotted on the graph
• of f.
• Notice that the domain values -1, 1, and 2 are associated with the
x axis and the range values f(-1) = 1, f(1) = 3, f(2) = 4 and are
associated with the y axis.
 3.1 Introduction to Limits
Estimating the Limit Numerically
Let f (x) = x + 2.
Watch the behavior of the values of f(x) when x is close to 2.
Then construct a table that shows values of f (x) for two sets
of x-values—one set that approaches 2 from the left and one
that approaches 2 from the right

x 1.9 1.99 1.999 2.0 2.001 2.01 2.1

f(x) 3.9 3.99 3.999 4 4.001 4.01 4.1
 3.1 Introduction to Limits
• The thin vertical lines in Figure 2 represent values of x that are
close to 2.
• The corresponding horizontal lines identify the value of
f(x) associated with each value of x.
• The graph indicates that as the values of x get closer and closer to
2 on either side of 2, the corresponding values of f(x) get closer
and closer to 4.
• Symbolically, we write
lim f (x)  4
x2
 3.1 Introduction to Limits
lim f (x)  4
x2

This equation is read as the limit of f(x) as x approaches 2 is 4.

Note that f(2) = 4.
That is, the value of the function at 2 and the limit of the function
as x
approaches 2 are the same.
lim This
f (x)relationship
 f (2)  can be expressed as
x2
4
Graphically, this means that there is no hole or
break in the graph of f
at x = 2.
 3.1 Introduction to Limits
DEFINITION Limit
We write
lim f (x)  L or f (x)  L as x 
xc
c
if the functional value f(x) is close to the single real number L
whenever x is close, but not equal, to c (on either side of c).
• Note: The existence of a limit at c has nothing to do with the
value of the function at c. In fact, c may not even be in the
domain of f. However, the function must be defined on both sides
of c.
 3.1 Introduction to Limits
Absolute value function

 x if x  0
f (x)  x   
x if x  0
 3.1 Introduction to Limits
Analyzing a Limit Let h(x) = |x|/x. Explore the behavior of h(x) for x
near, but not equal, to 0. Find lim x  0 h ( x ) if it exists
2 2
h(2)    1
2 2
0 0
h(0)   Not defined
0
20 2
h(2)  2  2  1
In general, h(x) is -1 for all negative x and 1 for all positive
x.
3.1 Introduction to Limits
Figure illustrates the behavior of h(x)
for x near 0. Note that the absence of a
solid dot on the vertical axis indicates
that h is not defined when x = 0
When x is near 0 (on either side of 0),
is h(x) near one specific number?
The answer is No, because h(x) is -1
for
x < 0 and 1 for x > 0. Consequently,
we say that
x
lim
x0 x does not exist
 3.1 Introduction to Limits
We see that the values of the function h(x) approached two
different numbers, depending on the direction of approach, and
it is natural to refer to these values as “the limit from the left”
and “the limit from the right”. These experiences suggest that
the notion of one-sided limits will be very useful in discussing
basic limit concepts.
 3.1 Introduction to Limits
DEFINITION One-Sided Limits
We write
lim f (c)  x →c - is read x approaches c from

xc the left and means x → c and x <

Kthe limit from the left orcthe left-hand limit if f(x) is close to
and call K
K whenever x is close to, but to the left of, c on the real number line.
We write
x →c+ is read x approaches c from
lim f (c)  the right and means x → c and x >
xc
and call LLthe limit from the right or
c the right-hand limit if f(x) is close
to L whenever x is close to, but to the right of, c on the real number line.
 3.1 Introduction to Limits
If no direction is specified in a limit statement, we will always assume
that the limit is two-sided or unrestricted. Theorem 1 states an
important relationship between one-sided limits and unrestricted limits.

THEOREM 1 On the Existence of a Limit

For a (two-sided) limit to exist, the limit from the left and the
limit from the right must exist and be equal. That is,

lim f (x)  L if and onlyif lim f (x)  lim f (x)

L
xc xc  xc 
 3.1 Introduction to Limits
In Example 3,

x x
lim x  1
x0  lim x 
x0 
and 1
Since the left- and right-hand limits are not the
same,
x
lim
x0 x does not exist
 3.1 Introduction to Limits
Analyzing Limits
Graphically Given the
graph of the function f in
Figure 5, discuss the
behavior of f(x) for x
near (A) -1 (B) 1, and
(C) 2.

,
 3.1 Introduction to Limits
(A)-1
For any x near
on either side of we see
that the corresponding
value of f(x), determined
by a horizontal line, is
close to 1

,
 3.1 Introduction to Limits
(B) 1
Again, for any x near, but not equal to, 1, the
vertical and horizontal lines indicate that the
corresponding value of f(x) is close to 3. The
open dot at (1, 3), together with the absence of
a solid dot anywhere on the vertical line
through x = 1, indicates that f(1) is not
defined.
 3.1 Introduction to Limits
(C)
The abrupt break in the graph at
indicates that the behavior of the
graph near is more complicated than
in the two preceding cases. If x is
close to 2 on the left side of 2, the
corresponding horizontal line
intersects the y axis at a point close to
2. If x is close to 2 on the right side of
2, the corresponding horizontal line
intersects the y axis at a point close to
5.This is a case where the one-sided
limits are different
 3.1 Introduction to Limits
In the example, note that lim
x1f (x) exists even though f is not defined

at x = 1 and the graph has a hole at x = 1. In general, the value of

a function at x = c has no effect on the limit of the function as x
approaches c.
 3.1 Introduction to Limits
Limits: An Algebraic Approach
Graphs are very useful tools for investigating limits, especially if
something unusual happens at the point in question. However, many
of the limits encountered in calculus are routine and can be evaluated
quickly with a little algebraic simplification, some intuition, and
basic properties of limits. The following list of properties of limits
forms the basis for this approach:
 3.1 Introduction to Limits
Using Limit Properties
1) Find lim(x2  4x)
x3

What happens if we try to evaluate a limit like the one , but with x
approaching an unspecified number, such as c?
2) Find lim(x  4x)
2
xc
3.1 Introduction to Limits
3.1 Introduction to Limits
 3.1 Introduction to Limits
Evaluating Limits Find each limit.
A) lim(x  5x 1)
3
x2

B) lim 2x  3 2
x1

2x
C) lim
x4 3x  1
 3.1 Introduction to Limits
Evaluating Limits Find each limit.

2 x  3x  2
2
Let f ( x)  2 , Find
A) lim f (x) x  x6
x2

B) lim f (x)
x0

C) lim f (x)
x1
 3.1 Introduction to Limits
Evaluating Limits Let
if x
f (x)   x 
2

1  2 if
 x

1 x2
A) lim f (x) B) lim f (x)
x2  x2 

C) lim f (x) D) f
x2
(2)
 3.1 Introduction to Limits
Evaluating Limits Let
if x  5
 2x  3
f (x)   if x  5
x
12
A) lim f (x) B) lim f (x)
x5 x5 

C) lim f (x) D) f
x5
(5)
 3.1 Introduction to Limits
• It is important to note that there are restrictions on some of the
limit properties. In particular, if
f (x)
xc f (x)  0 and lim
lim xc g(x)  0, then finding lim
xc g(x)

• may present some difficulties, since limit property 7 (the limit of

qa uotient) does not apply when lim g(x)  The next
x c
illustrates some techniques that 0c an be useful in this
example
situation.
 3.1 Introduction to Limits
• Evaluating Limits Find each limit
• Algebraic Simplification is often useful when the numerator
and denominator are both approaching 0.

x 4
2
x  4x 
2
A) B)
x2 3x  2 x3 x
lim lim
3
 3.1 Introduction to Limits
• Evaluating Limits Find each limit
• One-sided limits are helpful for limits involving the absolute
function
value xx
lim 1
x1 x 

1
 3.1 Introduction to Limits
DEFINITION Indeterminate
Form is said to be
•If limxcf (x)  0 and limxcg(x)  0, then limxc g(x)
f

(x)
indeterminate, or, more specifically, a 0/0 indeterminate form.
• The term indeterminate is used because the limit of
an indeterminate form may or may not exist.
• CAUTION The expression 0/0 does not represent a real number
and should never be used as the value of a limit. If a limit is a
0/0 indeterminate form, further investigation is always required
to determine whether the limit exists and to find its value if it
does exist.
 3.1 Introduction to Limits
• If the denominator of a quotient approaches 0 and the numerator
approaches a nonzero number, then the limit of the quotient is
not an indeterminate form. In fact, a limit of this form never
exists.
• THEOREM 4 Limit of a Quotient
If lim f (x)  L, L  0, and lim g(x)  0,
xc xc

then
f (x)
lim
xc g(x) does not

exist
 3.1 Introduction to Limits
Limits of Difference Quotients
• Let the function f be defined in an open interval containing the
number a. One of the most important limits in calculus is the
limit of the difference quotient,
f (a  h)  f (a)
lim
xc h
If

h0  f (a  h) 
lim
• As it often does, then limit is an indeterminate
f (a)  0
 3.1 Introduction to Limits
Limits of Difference Quotients
1) Find the following limit for f(x) = 4x – 5:

lim
h0 h
2) Find the following limit for f(x) = 7 – 2x:
f (3  h) 
lim h
f (3)
h0
 3.1 Introduction to Limits
Limits of Difference Quotients
1) Find the following limit for f(x) = |x + 5|:

lim
h0 h

2) Find the following limit for x :

f (5  h) 
f (2  h)  f (2)
flim
(5)
h0 h
 Practice
Ex. 3.1 (pg. 128)
#’s 1, 3, 5, 7, 19, 29, 37, 41, 47, 53