Question

Evaluate the limit if it exists.

$\underset{x\to 7}{\lim }\frac{\sqrt{x+2}-3}{x-7}$

Short answer

The value of limit of the given function is$0.$

Step-by-step solution

Step 1. Given information.

The limit of function is given here,

$\underset{x\to 7}{\lim }\frac{\sqrt{x+2}-3}{x-7}$

Step 2. Formula used.

The given function is a rational number and $x$is in its domain so we can find the limit by direct substitution.

Step 3. Simplification.

Given,

$\begin{array}{c}\lim \frac{\sqrt{x+2}-3}{x-7}=\frac{\sqrt{7+2}-3}{7-7}\\ =\frac{\sqrt{9}-3}{0}\\ =\frac{3-3}{0}\\ =0\end{array}$

Therefore, our final answer is$0.$