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Chapter 2 | Limits
Finding a Vertical Asymptote
Evaluate each of the following limits using Infinite Limits from Positive Integers . Identify any vertical asymptotes of the function f ( x ) =1/( x +3) 4 .
1 ( x +3) 4 1 ( x +3) 4
lim x →−3 −
a.
lim x →−3 +
b.
1 ( x +3) 4
lim x →−3
c.
Solution We can use Infinite Limits from Positive Integers directly. a. lim x →−3 − 1 ( x +3) 4 =+∞ b. lim x →−3 + 1 ( x +3) 4 =+∞ c. lim x →−3 1 ( x +3) 4 =+∞ The function f ( x ) =1/( x +3) 4 has a vertical asymptote of x =−3.
2.9
Evaluate each of the following limits. Identify any vertical asymptotes of the function f ( x ) = 1 ( x −2) 3 .
1 ( x −2) 3 1 ( x −2) 3
lim x →2 −
a.
lim x →2 +
b.
1 ( x −2) 3
c. lim x →2
In the next example we put our knowledge of various types of limits to use to analyze the behavior of a function at several different points. Example 2.11 Behavior of a Function at Different Points
Use the graph of f ( x ) in Figure 2.21 to determine each of the following values: a. lim x →−4 − f ( x ); lim x →−4 + f ( x ); lim x →−4 f ( x ); f (−4)
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