Chapter 11.3, Problem 55E

Solution

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

  $\underset{x\to 3^{-}}{\lim }\frac{1}{x-3}=-\infty$

Vertical asymptote $x=3$ .

Given information:

The given function is $f\left(x\right)=\frac{1}{x-3}$ .

Calculation:

Consider the function $f\left(x\right)=\frac{1}{x-3}$ .

The graph of this function is

  

Figure (1) The graph suggests that

  $\underset{x\to 3^{-}}{\lim }\frac{1}{x-3}=-\infty$

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

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

$x$	$f\left(x\right)=\frac{1}{x-3}$
$2.9$	$-10$
$2.99$	$-100$
$2.999$	$-1000$
$2.9999$	$-10000$
$2$	error

Thus, $\underset{x\to 3^{-}}{\lim }\frac{1}{x-3}=-\infty$ and this function has vertical asymptote $x=3$ .