Chapter 11.3, Problem 56E

Solution

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

  $\underset{x\to 3^{+}}{\lim }\frac{x}{x-3}=-\infty$

Vertical asymptote $x=3$ .

Given information:

The given function is $f\left(x\right)=\frac{x}{x-3}$ .

Calculation:

Consider the function $f\left(x\right)=\frac{x}{x-3}$ .

The graph of this function is

  

Figure (1) The graph suggests that

  $\underset{x\to 3^{+}}{\lim }\frac{x}{x-3}=\infty$

And the given function has vertical asymptote $x=3$

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

$x$	$f\left(x\right)=\frac{x}{x-3}$
$3$	Error
$3.00001$	$300001$
$3.0001$	$30001$
$3.001$	$3001$
$3.01$	$301$

Thus, $\underset{x\to 3^{+}}{\lim }\frac{x}{x-3}=\infty$ and this function has vertical asymptote $x=3$ .