1.8: Infinite Limits
Learning Objectives
A student will be able to:
 Find infinite limits of functions.
 Analyze properties of infinite limits.
 Identify asymptotes of functions.
 Analyze end behavior of functions.
Introduction
In this lesson we will discuss infinite limits. In our discussion the notion of infinity is discussed in two contexts. First, we can discuss infinite limits in terms of the value a function as we increase
The second context in which we speak of infinite limits involves situations where the function values increase without bound. For example, in the case of a rational function such as
At
In this example we note that
More formally, we define these as follows:
Definition:

The righthand limit of the function
f(x) atx=a is infinite, and we writelimx→a+f(x)=∞ , if for every positive numberk , there exists an open interval(a,a+δ) contained in the domain off(x) , such thatf(x) is in(k,∞) for everyx in(a,a+δ) .  The definition for negative infinite limits is similar.

Suppose we look at the function
f(x)=(x+1)/(x2−1) and determine the infinite limitslimx→∞f(x) andlimx→−∞f(x) . 
We observe that as
x increases in the positive direction, the function values tend to get smaller. The same is true if we decreasex in the negative direction. Some of these extreme values are indicated in the following table.

We observe that the values are getting closer to
f(x)=0. Hencelimx→∞f(x)=0 andlimx→−∞f(x)=0 . 
Since our original function was roughly of the form
f(x)=1x , this enables us to determine limits for all other functions of the formf(x)=1xp withp>0. Specifically, we are able to conclude thatlimx→∞1xp=0 . This shows how we can find infinite limits of functions by examining the end behavior of the functionf(x)=1xp,p>0.  The following example shows how we can use this fact in evaluating limits of rational functions.
Example 1:
Find
Solution:
Note that we have the indeterminate form, so Limit Property #5 does not hold. However, if we first divide both numerator and denominator by the quantity
We observe that the limits
Lesson Summary
 We learned to find infinite limits of functions.
 We analyzed properties of infinite limits.
 We identified asymptotes of functions.
 We analyzed end behavior of functions.
Multimedia Links
For more examples of limits at infinity (1.0), see Math Video Tutorials by James Sousa, Limits at Infinity (9:42).
Review Questions
In problems 1  7, find the limits if they exist.

limx→3+(x+2)2(x−2)2−1 
limx→∞(x+2)2(x−2)2−1  \begin{align*}\lim_{x \to 1^+} \frac{(x + 2)^2} {(x  2)^2  1}\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{2x  1} {x + 1}\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{x^5 + 3x^4 + 1} {x^3  1}\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{3x^4  2x^2 + 3x + 1} {2x^4  2x^2 + x  3}\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{2x^2  x + 3} {x^5  2x^3 + 2x  3}\end{align*}
In problems 8  10, analyze the given function and identify all asymptotes and the end behavior of the graph.
 \begin{align*}f(x) = \frac{(x + 4)^2} {(x  4)^2  1}\end{align*}
 \begin{align*}f(x) = 3x^3  x^2 + 2x + 2\end{align*}
 \begin{align*}f(x) = \frac{2x^2  8} {x + 2}\end{align*}
 Consider \begin{align*}f(x) = \frac{1} {x + 1}, g(x) = x^2\end{align*}. We previously found \begin{align*}\lim_{x \to 1} (f \circ g) (x) = \frac{1} {2}\end{align*}. Find \begin{align*}\lim_{x \to 1} (g \circ f)(x).\end{align*}
Review Answers
 \begin{align*}\lim_{x \to 3^+} \frac{(x + 2)^2} {(x  2)^2  1} = +\infty\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{(x + 2)^2} {(x  2)^2  1}=1\end{align*}
 \begin{align*}\lim_{x \to 1^+} \frac{(x + 2)^2} {(x  2)^2  1} =  \infty\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{2x  1} {x + 1} = 2\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{x^5 + 3x^4 + 1} {x^3  1} = \infty\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{3x^4  2x^2 + 3x + 1} {2x^4  2x^2 + x  3} = \frac{3} {2}\end{align*}
 \begin{align*}\lim_{x \to \infty} \frac{2x^2  x + 3} {x^5  2x^3 + 2x  3} = 0\end{align*}
 Zero at \begin{align*}x = 4\end{align*}; vertical asymptotes at \begin{align*}x = 3, 5\end{align*}; \begin{align*}f(x) \rightarrow 1\end{align*} as \begin{align*}x \rightarrow \pm \infty.\end{align*}
 Zero at \begin{align*}x = 1\end{align*}; no vertical asymptotes; \begin{align*}f(x) \rightarrow \infty\end{align*} as \begin{align*}x \rightarrow \infty\end{align*}; \begin{align*}f(x) \rightarrow \infty\end{align*} as \begin{align*}x \rightarrow \infty.\end{align*}
 Zero at \begin{align*}x = 2\end{align*}; no vertical asymptotes but there is a discontinuity at \begin{align*}x = 2\end{align*}; \begin{align*}f(x) \rightarrow \infty\end{align*} as \begin{align*}x \rightarrow \infty\end{align*}; \begin{align*} f(x) \rightarrow \infty\end{align*} as \begin{align*}x \rightarrow \infty.\end{align*}
 \begin{align*}\lim_{x \to 1} (g \circ f)(x)= +\infty\end{align*}
Texas Instruments Resources
In the CK12 Texas Instruments Calculus FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9726.
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