Limits at Infinity and Asymptotes: Learn It 5

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Limits at Infinity and Asymptotes: Learn It 5

End Behavior Cont.

Determining End Behavior for Transcendental Functions

Trigonometric functions

The six basic trigonometric functions are periodic and do not approach a finite limit as $x\to \pm \infty$ .

For example, $\sin x$ oscillates between 1 and -1 (Figure 19). The tangent function $x$ has an infinite number of vertical asymptotes as $x\to \pm \infty$ ; therefore, it does not approach a finite limit nor does it approach $\pm \infty$ as $x\to \pm \infty$ as shown in (Figure 20).

The function f(x) = sin x is graphed. — Figure 19. The function $f(x)= \sin x$ oscillates between 1 and -1 as $x\to \pm \infty$

The function f(x) = tan x is graphed. — Figure 20. The function $f(x)= \tan x$ does not approach a limit and does not approach $\pm \infty$ as $x\to \pm \infty$

Exponential functions

Recall that for any base $b>0, \, b\ne 1$ , the function $y=b^x$ is an exponential function with domain $(−\infty ,\infty )$ and range $(0,\infty )$ . If $b>1, \, y=b^x$ is increasing over $(−\infty ,\infty )$ .If [latex]0

For the natural exponential function $f(x)=e^x$ , $e\approx 2.718>1$ . Therefore, $f(x)=e^x$ is increasing on $(−\infty ,\infty )$ and the range is $(0,\infty)$ . The exponential function $f(x)=e^x$ approaches $\infty$ as $x\to \infty$ and approaches 0 as $x\to −\infty$ .

End behavior of the natural exponential function
$x$	$-5$	$-2$	$0$	$2$	$5$
$e^x$	$0.00674$	$0.135$	$1$	$7.389$	$148.413$

The function f(x) = ex is graphed. — Figure 21. The exponential function approaches zero as $x\to −\infty$ and approaches $\infty$ as $x\to \infty$ .

Recall that the natural logarithm function $f(x)=\ln (x)$ is the inverse of the natural exponential function $y=e^x$ . Therefore, the domain of $f(x)=\ln (x)$ is $(0,\infty )$ and the range is $(−\infty ,\infty )$ .

The graph of $f(x)=\ln (x)$ is the reflection of the graph of $y=e^x$ about the line $y=x$ . Therefore, $\ln (x)\to −\infty$ as $x\to 0^+$ and $\ln (x)\to \infty$ as $x\to \infty$ .

End behavior of the natural logarithm function
$x$	$0.01$	$0.1$	$1$	$10$	$100$
$\ln (x)$	$-4.605$	$-2.303$	$0$	$2.303$	$4.605$

The function f(x) = ln(x) is graphed. — Figure 22. The natural logarithm function approaches $\infty$ as $x\to \infty$ .

Find the limits as $x\to \infty$ and $x\to −\infty$ for $f(x)=\frac{(2+3e^x)}{(7-5e^x)}$ and describe the end behavior of $f$ .

To find the limit as $x\to \infty$ , divide the numerator and denominator by $e^x$ :

\begin{matrix} lim x \to \infty f (x) & = lim x \to \infty \frac{2 + 3 e^{x}}{7 - 5 e^{x}} = lim x \to \infty \frac{(2 / e^{x}) + 3}{(7 / e^{x}) - 5} . \end{matrix}

$\begin{array}{ll} \underset{x\to \infty }{\lim}f(x) & =\underset{x\to \infty }{\lim}\frac{2+3e^x}{7-5e^x} \\ & =\underset{x\to \infty }{\lim}\frac{(2/e^x)+3}{(7/e^x)-5}. \end{array}$

As shown in Figure 21, $e^x\to \infty$ as $x\to \infty$ . Therefore,

lim x \to \infty \frac{2}{e^{x}} = 0 = lim x \to \infty \frac{7}{e^{x}}

$\underset{x\to \infty }{\lim}\frac{2}{e^x}=0=\underset{x\to \infty }{\lim}\frac{7}{e^x}$ .

We conclude that $\underset{x\to \infty }{\lim}f(x)=-\frac{3}{5}$ , and the graph of $f$ approaches the horizontal asymptote $y=-\frac{3}{5}$ as $x\to \infty$ .

To find the limit as $x\to −\infty$ , use the fact that $e^x \to 0$ as $x\to −\infty$ to conclude that $\underset{x\to -\infty }{\lim}f(x)=\frac{2}{7}$ , and therefore the graph of approaches the horizontal asymptote $y=\frac{2}{7}$ as $x\to −\infty$ .

Watch the following video to see the worked solution to this example.

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You can view the transcript for this segmented clip of “4.6 Limits at Infinity and Asymptotes” here (opens in new window).