Chapter 11.3, Problem 58E

Solution

To find: Limit and vertical asymptotes of the given function using graph and tables.

  $\underset{x\to -2^{-}}{\lim }\frac{x}{x+2}=-\infty$

Vertical asymptote $x=-2$ .

Given information:

The given function is $f\left(x\right)=\frac{x}{x+2}$ .

Calculation:

Consider the function $f\left(x\right)=\frac{x}{x+2}$ .

The graph of this function is

  

Figure (1) The graph suggests that

  $\underset{x\to -2^{-}}{\lim }\frac{x}{x+2}=-\infty$

And the given function has vertical asymptote $x=-2$

The table explaining the limit of the given function is as follows

$x$	$f\left(x\right)=\frac{x}{x+2}$
$-2.0001$	$200001$
$-2.0001$	$20001$
$-2.001$	$2001$
$-2.01$	$201$
$-2$	Error

Thus, $\underset{x\to -2^{+}}{\lim }\frac{1}{x+2}=\infty$ and this function has vertical asymptote $x=-2$ .