Chapter 11.1, Problem 48E

To determine

Graph the function $f\left(x\right)=\frac{\sqrt{x+5}-4}{x-2},\underset{x\to 2}{\lim }f\left(x\right)$ and determine whether the limit exists.

Answer to Problem 48E

The limit does not exist.

Explanation of Solution

Given:

  $f\left(x\right)=\frac{\sqrt{x+5}-4}{x-2},\underset{x\to 2}{\lim }f\left(x\right)$

Graph:

By using a graphing utility, the graph of given function is

  

Interpretation:

The graph shows that as $x$ approaches 2 from the right, $f\left(x\right)$ approaches $-\infty$ and as $x$ approaches 2 from the left, $f\left(x\right)$ approaches $+\infty$ .

The limit does not exist because $\underset{x\to a-}{\lim }f\left(x\right)\ne \underset{x\to a+}{\lim }f\left(x\right)$ and the sum diverges at 2.

Conclusion:

Therefore, the limit does not exist as $x$ approaches 2.