4.6 Limits at infinity and asymptotes (Page 2/14)

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The function f(x) = sin x is graphed. — The function $f (x) = sin x$ oscillates between $1 and −1$ as $x \to ± \infty$

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

Recall that for any base $b > 0, b \neq 1,$ 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 $0 < b < 1,$ $y = b^{x}$ is decreasing over $(− \infty, \infty) .$ 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 $x \to − \infty$ as shown in [link] and [link] .

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. — 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 $x \to \infty$ as shown in [link] and [link] .

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. — The natural logarithm function approaches $\infty$ as $x \to \infty .$

Determining end behavior for a transcendental function

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

As shown in [link] , $e^{x} \to \infty$ as $x \to \infty .$ Therefore,

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

We conclude that $lim_{x \to \infty} f (x) = - \frac{3}{5},$ 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 $lim_{x \to \infty} f (x) = \frac{2}{7},$ and therefore the graph of approaches the horizontal asymptote $y = \frac{2}{7}$ as $x \to − \infty .$

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Find the limits as $x \to \infty$ and $x \to − \infty$ for $f (x) = \frac{(3 e^{x} - 4)}{(5 e^{x} + 2)} .$

$lim_{x \to \infty} f (x) = \frac{3}{5},$ $lim_{x \to − \infty} f (x) = −2$

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Guidelines for drawing the graph of a function

We now have enough analytical tools to draw graphs of a wide variety of algebraic and transcendental functions. Before showing how to graph specific functions, let’s look at a general strategy to use when graphing any function.

Problem-solving strategy: drawing the graph of a function

Given a function $f,$ use the following steps to sketch a graph of $f :$

Determine the domain of the function.
Locate the $x$ - and $y$ -intercepts.
Evaluate $lim_{x \to \infty} f (x)$ and $lim_{x \to − \infty} f (x)$ to determine the end behavior. If either of these limits is a finite number $L,$ then $y = L$ is a horizontal asymptote. If either of these limits is $\infty$ or $− \infty,$ determine whether $f$ has an oblique asymptote. If $f$ is a rational function such that $f (x) = \frac{p (x)}{q (x)},$ where the degree of the numerator is greater than the degree of the denominator, then $f$ can be written as

$f (x) = \frac{p (x)}{q (x)} = g (x) + \frac{r (x)}{q (x)},$

where the degree of $r (x)$ is less than the degree of $q (x) .$ The values of $f (x)$ approach the values of $g (x)$ as $x \to ± \infty .$ If $g (x)$ is a linear function, it is known as an oblique asymptote .
Determine whether $f$ has any vertical asymptotes.
Calculate $f^{'} .$ Find all critical points and determine the intervals where $f$ is increasing and where $f$ is decreasing. Determine whether $f$ has any local extrema.
Calculate $f ″ .$ Determine the intervals where $f$ is concave up and where $f$ is concave down. Use this information to determine whether $f$ has any inflection points. The second derivative can also be used as an alternate means to determine or verify that $f$ has a local extremum at a critical point.

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Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

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