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Recall that for any base $b>0,b\ne 1,$ the function $y={b}^{x}$ is an exponential function with domain $\left(\text{\u2212}\infty ,\infty \right)$ and range $\left(0,\infty \right).$ If $b>1,y={b}^{x}$ is increasing over $`\left(\text{\u2212}\infty ,\infty \right).$ If $0<b<1,$ $y={b}^{x}$ is decreasing over $\left(\text{\u2212}\infty ,\infty \right).$ For the natural exponential function $f\left(x\right)={e}^{x},$ $e\approx 2.718>1.$ Therefore, $f\left(x\right)={e}^{x}$ is increasing on $`\left(\text{\u2212}\infty ,\infty \right)$ and the range is $`\left(0,\infty \right).$ The exponential function $f\left(x\right)={e}^{x}$ approaches $\infty $ as $x\to \infty $ and approaches $0$ as $x\to \text{\u2212}\infty $ as shown in [link] and [link] .
$x$ | $\mathrm{-5}$ | $\mathrm{-2}$ | $0$ | $2$ | $5$ |
${e}^{x}$ | $0.00674$ | $0.135$ | $1$ | $7.389$ | $148.413$ |
Recall that the natural logarithm function $f\left(x\right)=\text{ln}\left(x\right)$ is the inverse of the natural exponential function $y={e}^{x}.$ Therefore, the domain of $f\left(x\right)=\text{ln}\left(x\right)$ is $\left(0,\infty \right)$ and the range is $\left(\text{\u2212}\infty ,\infty \right).$ The graph of $f\left(x\right)=\text{ln}\left(x\right)$ is the reflection of the graph of $y={e}^{x}$ about the line $y=x.$ Therefore, $\text{ln}\left(x\right)\to \text{\u2212}\infty $ as $x\to {0}^{+}$ and $\text{ln}\left(x\right)\to \infty $ as $x\to \infty $ as shown in [link] and [link] .
$x$ | $0.01$ | $0.1$ | $1$ | $10$ | $100$ |
$\text{ln}\left(x\right)$ | $\mathrm{-4.605}$ | $\mathrm{-2.303}$ | $0$ | $2.303$ | $4.605$ |
Find the limits as $x\to \infty $ and $x\to \text{\u2212}\infty $ for $f\left(x\right)=\frac{\left(2+3{e}^{x}\right)}{\left(7-5{e}^{x}\right)}$ and describe the end behavior of $f.$
To find the limit as $x\to \infty ,$ divide the numerator and denominator by ${e}^{x}\text{:}$
As shown in [link] , ${e}^{x}\to \infty $ as $x\to \infty .$ Therefore,
We conclude that $\underset{x\to \infty}{\text{lim}}f\left(x\right)=-\frac{3}{5},$ and the graph of $f$ approaches the horizontal asymptote $y=-\frac{3}{5}$ as $x\to \infty .$ To find the limit as $x\to \text{\u2212}\infty ,$ use the fact that ${e}^{x}\to 0$ as $x\to \text{\u2212}\infty $ to conclude that $\underset{x\to \infty}{\text{lim}}f\left(x\right)=\frac{2}{7},$ and therefore the graph of approaches the horizontal asymptote $y=\frac{2}{7}$ as $x\to \text{\u2212}\infty .$
Find the limits as $x\to \infty $ and $x\to \text{\u2212}\infty $ for $f\left(x\right)=\frac{\left(3{e}^{x}-4\right)}{\left(5{e}^{x}+2\right)}.$
$\underset{x\to \infty}{\text{lim}}f\left(x\right)=\frac{3}{5},$ $\underset{x\to \text{\u2212}\infty}{\text{lim}}f\left(x\right)=\mathrm{-2}$
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.
Given a function $f,$ use the following steps to sketch a graph of $f\text{:}$
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