lim
x
→
2
g x
,
lim
x
→
2
f x
,
lim
x
→
c
1
f x
lim
x
→
c
5
g x
4
f x
lim
x
→
c
6
f x g x
lim
x
→
c
f x
g x
2
lim
x
→
c
g x
2
lim
x
→
c
f x
5,
lim
x
→
c
f x
lim
x
→
c
f x
g x
lim
x
→
c
f x
g x
lim
x
→
c
2
g x
lim
x
→
c
g x
6
lim
x
→
c
f x
3,
lim
x
→
1
f x
f x
ln 7
x
,
lim
x
→
4
f x
f x
ln
x
3 ,
lim
x
→
3
f x
f x
7
x
3
,
lim
x
→
1
f x
f x
x
1
x
2
4
x
3
,
lim
x
→
2
f x
f x
x
5
4
x
2
,
lim
x
→
4
f x
f x
x
3
1
x
4
,
lim
x
→
1
f x
f x
sin
x
,
lim
x
→
0
f x
f x
cos
1
x
,
lim
x
→
0
f x
f x
e
x
1
x
,
lim
x
→
0
f x
f x
5
2
e
1
x
,

11.2
Techniques for Evaluating Limits
What
you
should
learn
Use the dividing out technique to evaluate
limits of functions.
Use the rationalizing technique to
evaluate limits of functions.
Approximate limits of functions
graphically and numerically.
Evaluate one-sided limits of functions.
Evaluate limits of difference quotients
from calculus.
Why
you
should
learn
it
Many definitions in calculus involve the
limit of a function. For instance, in Exercises
77 and 78 on page 799, the definition of the
velocity of a free-falling object at any instant
in time involves finding the limit of a
position function.
Section 11.2
Techniques for Evaluating Limits
791
Dividing Out Technique
In Section 11.1, you studied several types of functions whose limits can be
evaluated by direct substitution. In this section, you will study several techniques
for evaluating limits of functions for which direct substitution fails.
Suppose you were asked to find the following limit.
Direct substitution fails because
is a zero of the denominator. By using a
table, however, it appears that the limit of the function as
is
Another way to find the limit of this function is shown in Example 1.
5.
x
→
3
3
lim
x
→
3
x
2
x
6
x
3
Example 1
Dividing Out Technique
Find the limit:
Solution
Begin by factoring the numerator and dividing out any common factors.
Factor numerator.
Divide out common factor.
Simplify.
Direct substitution
Simplify.
Now try Exercise 7.
5
3
2
lim
x
→
3
x
2
lim
x
→
3
x
2
x
3
x
3
lim
x
→
3
x
2
x
6
x
3
lim
x
→
3
x
2
x
3
x
3
lim
x
→
3
x
2
x
6
x
3
.
This procedure for evaluating a limit is called the
dividing out technique.
The validity of this technique stems from the fact that if two functions agree at all
but a single number
c
, they must have identical limit behavior at
In
Example 1, the functions given by
and
agree at all values of
other than
So, you can use
to find the limit
of
f x
.
g x
x
3.
x
g x
x
2
f x
x
2
x
6
x
3
x
c
.
Peticolas Megna/Fundamental Photographs
x
3.01
3.001
3.0001
3
2.9999
2.999
2.99
x
2
x
6
x
3
5.01
5.001
5.0001
?
4.9999
4.999
4.99
Prerequisite Skills
To review factoring techniques, see
Appendix G, Study Capsule 1.

The dividing out technique should be applied only when direct substitution
produces 0 in both the numerator
and
the denominator. The resulting fraction,
has no meaning as a real number. It is called an
indeterminate form
because you
cannot, from the form alone, determine the limit. When you try to evaluate a limit
of a rational function by direct substitution and encounter this form, you can
conclude that the numerator and denominator must have a common factor. After

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