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Activity9.6End Behavior

We are frequently interested in a function's “end behavior.” That is, what is the behavior of the function as the input variable increases without bound or decreases without bound.

Many times a function will approach a horizontal asymptote as its end behavior. If the horizontal asymptote \(y=L\) represents the end behavior of the function \(f\) both as \(x\) increases without bound and as \(x\) decreases without bound, we write \(\lim\limits_{x\to\infty}\fe{f}{x}=L\) and \(\lim\limits_{x\to-\infty}\fe{f}{x}=L\).

While working Limits and Continuity you investigated strategies for formally establishing limit values as \(x\to\infty\) or \(x\to-\infty\). In this activity you are going to investigate a more informal strategy for determining these type limits.

Consider \(\lim\limits_{x\to\infty}\frac{4x-2}{3+20x}\). When the value of \(x\) is really large, we say that the term \(4x\) dominates the numerator of the expression and the term \(20x\) dominates the denominator. We actually call those terms the dominant terms of the numerator and denominator. The dominant terms are significant because when the value of \(x\) is really large, the other terms in the expression contribute almost nothing to the value of the expression. That is, for really large values of \(x\):

\begin{align*} \frac{4x-2}{3+20x}&\approx\frac{4x}{20x}\\ &=\frac{1}{5} \end{align*}

For example, even if \(x\) has the paltry value of \(1000\), \begin{align*} \frac{4(1000)-2}{3+20(1000)}&=\frac{3998}{20003}\\ &\approx0.19987\ldots\approx\frac{1}{5} \end{align*}

This tells us that \(\lim\limits_{x\to\infty}\frac{4x-2}{3+20x}=\frac{1}{5}\) and that \(y=\frac{1}{5}\) is a horizontal asymptote for the graph of \(y=\frac{4x-2}{3+20x}\).

Subsection9.6.1Exercises

The formulas used to graph Figures 9.6.19.6.5 are given below. Focusing first on the dominant terms of the expressions, match the formulas with the functions (\(f_1\) through \(f_5\)).

1

\(y=\dfrac{3x+6}{x-2}\)

2

\(y=\dfrac{16+4x}{6+x}\)

3

\(y=\dfrac{6x^2-6x-36}{36-3x-3x^2}\)

4

\(y=\dfrac{-2x+8}{x^2-100}\)

5

\(y=\dfrac{15}{x-5}\)

Figure9.6.1\(y=\fe{f_1}{x}\)
Figure9.6.2\(y=\fe{f_2}{x}\)
Figure9.6.3\(y=\fe{f_3}{x}\)
Figure9.6.4\(y=\fe{f_4}{x}\)
Figure9.6.5\(y=\fe{f_5}{x}\)

Use the concept of dominant terms to informally determine the value of each of the following limits.

6

\(\lim\limits_{x\to-\infty}\dfrac{4+x-7x^3}{14x^3+x^2+2}\)

7

\(\lim\limits_{t\to-\infty}\dfrac{4t^2+1}{4t^3-1}\)

8

\(\lim\limits_{\gamma\to\infty}\dfrac{8}{2\gamma^3}\)

9

\(\lim\limits_{x\to\infty}\dfrac{(3x+1)(6x-2)}{(4+x)(1-2x)}\)

10

\(\lim\limits_{t\to\infty}\dfrac{4e^t-8e^{-t}}{e^t+e^{-t}}\)

11

\(\lim\limits_{t\to-\infty}\dfrac{4e^t-8e^{-t}}{e^t+e^{-t}}\)