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Limits and Their Properties11.2
Copyright © Cengage Learning. All rights reserved.
2
2
2
lim
lim 5
3)
2
(2
x
x
f x
f x
f
2
limxf x
DNE
Finding a limit graphically
Warm-up:
Evaluating Limits Analytically
Copyright © Cengage Learning. All rights reserved.
1.2
4
Evaluate a limit using properties of limits.
Develop and use a strategy for finding limits.
Evaluate a limit using dividing out and rationalizing techniques.
Evaluate a limit using the Squeeze Theorem.
Objectives
5
Properties of Limits
6
The limit of f (x) as x approaches c does not depend on the value of f at x = c. It may happen, however, that the limit is precisely f (c).
In such cases, the limit can be evaluated by direct substitution. That is,
Such well-behaved functions are continuous at c.
Properties of Limits
7
Example 1 – Evaluating Basic Limits
8
Example 2 – Evaluating Basic Limits
2
2lim 4 3x
Evaluate x
9
The limit (as x → 2 ) of the polynomial function
p(x) = 4x2 + 3 is simply the value of p at x = 2.
This direct substitution property is valid for all polynomial
and rational functions with nonzero denominators.
Properties of Limits
10
Properties of Limits
In other words, use direct substitution unless it results in division by zero.
11
Find the limit:
Solution:
Because the denominator is not 0 when x = 1, you can apply Theorem 1.3 to obtain
Example 3 – The Limit of a Rational Function
13
Example 4– The Limit of a Composite Function
Find the limit:
14
Example 5 – Limits of Trigonometric Functions
15
A Strategy for Finding Limits
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A Strategy for Finding Limits
17
Find the limit:
Example 6 – Finding the Limit of a Function
18
Example 6 – Solution
So, for all x-values other than x = 1, the functions f and g agree, as shown in Figure 1.17
Figure 1.17
cont’d
f and g agree at all but one point
19
A Strategy for Finding Limits
20
Dividing Out and Rationalizing Techniques
21
Dividing Out and Rationalizing Techniques
Two techniques for finding limits analytically are
shown in Examples 7 and 8.
The dividing out technique involves dividing out
common factors, and the rationalizing technique
involves rationalizing the numerator of a fractional
expression.
22
Example 7 – Dividing Out Technique
Find the limit:
Solution:
Although you are taking the limit of a rational function, you cannot apply Theorem 1.3 because the limit of the denominator is 0.
23
Because the limit of the numerator is also 0, the numerator and denominator have a common factor of (x + 3).
So, for all x ≠ –3, you can divide out this factor to obtain
Using Theorem 1.7, it follows that
Example 7 – Solutioncont’d
24
This result is shown graphically in Figure 1.18.
Note that the graph of the function f coincides with the graph of the function g(x) = x – 2, except that the graph of f has a gap at the point (–3, –5).
Example 7 – Solution
Figure 1.18
cont’d
25
2
21
2 3Evaluate lim .
1x
x x
x
Theorem
You Try:
26
You Try:
3
3
27Evaluate lim
3x
x
x
27
An expression such as 0/0 is called an indeterminate form because you cannot (from the form alone) determine the limit.
When you try to evaluate a limit and encounter this form, remember that you must rewrite the fraction so that the new denominator does not have 0 as its limit.
One way to do this is to divide out like factors, as shown in Example 7. A second way is to rationalize the numerator, as shown in Example 8.
Dividing Out and Rationalizing Techniques
28
Find the limit:
Solution:
By direct substitution, you obtain the indeterminate form 0/0.
Example 8 – Rationalizing Technique
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In this case, you can rewrite the fraction by rationalizing the numerator.
cont’d Example 8 – Solution
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Now, using Theorem 1.7, you can evaluate the limit as shown.
cont’d Example 8 – Solution
31
A table or a graph can reinforce your conclusion that the
limit is . (See Figure 1.20.)
Figure 1.20
Example 8 – Solutioncont’d
32
Example 8 – Solutioncont’d
33
You Try:
0
4 2Evaluate lim
x
x
x
34
You Try:
0
1 1
2 2Evaluate limx
xx
35
Example:
0
2
Evaluate lim
1, 0
1, 0
x
f x
x xf x
x x
0
limx
f x
0
limx
f x
1 1
36
0
lim 1xf x
37
You Try:
0
2Evaluate lim
x
x
x
limDNE
38
You Try:
2
Evaluate lim
8, 0 1
10, 1 2
12, 2 3
x
f x
x
f x x
x
limDNE
39
You Try:
1
3Evaluate lim
x
x
x
lim 3
40
Limit of a Difference Quotient:
2
0
For the function 1,
3 3limh
f x x find
f h f
h
0lim 6h
41
You Try:
2
0
For the function 2 3,
4 4limh
f x x x find
f h f
h
0lim 10h
43
Homework
Day 1 11.2 pg. 760 1-29 odd, 43-49 odd.
Day 2: 11.2 pg. 760 2-24 even, 59-65 odd
44
Mini Quiz 4/24
9
9
3x
xLim
x
1) Find the limit analytically:
45
Mini Quiz 4/24
1
1 12 1 3
1x
xLimx
2) Find the limit algebraically:
46
Mini Quiz 4/24
0
sec
tanx
xLim
x
2) Find the limit analytically:
