Question

Sketch careful, labeled graphs of each function $f$in Exercises 57-82 by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$and $f^{\text{'}}$, and examine any relevant limits so that you can describe all key points and behaviors of $f$.

$f\left(x\right)=e^{\ln e^x}$

Short answer

The function $f$is defined everywhere; it has no roots; the function is positive everywhere. There is no local extremum. The function is increasing on $(-\infty ,\infty )$and $\underset{x\to -\infty }{\lim }f\left(x\right)=0$, so there is a horizontal asymptote on the left at $y=0$. $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$, so there is no horizontal asymptote on the right.

Step-by-step solution

Given information

The given function is $f\left(x\right)=e^{\ln e^n}$

Calculation

Let us consider

$f\left(x\right)=e^{\ln e^x}$

The table values

$x$	$-2$	-1	$0$	1
$f\left(x\right)$	$0.14$	$0.37$	$1$	$e$

By equating the first derivative to zero, determine the critical spots.

$$\begin{gathered} f\left(x\right)=e^{\ln e^x} \\ {f}^{\text{'}}\left(x\right)=e^{\ln e^x}\frac{1}{e^x}=0 \end{gathered}$$

There are no local extrema and no critical spots.

The derivative's sign diagram:

Find the limits of the function

localid="1658379728716" $$\begin{gathered} \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }{e}^{\ln e^x}=\infty \\ \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }{e}^{\ln e^x}=0 \end{gathered}$$

Therefore, the function is increasing onlocalid="1658379740081" $(-\infty ,\infty )$. There are no local maxima or minima.

Everywhere defines the function and it has none at all.

The limits of the function localid="1658379702150" $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$and localid="1658379720032" $\underset{x\to -\infty }{\lim }f\left(x\right)=0$.

Draw the graph by hand as follows