Chapter 12.2, Problem 7E

To determine

ToDetermine:The limit $\underset{x\to 6}{\lim }\left(\frac{x^2-36}{x-6}\right)$ , if it exists. Use a graphing utility to verify the result graphically.

Answer to Problem 7E

The $\underset{x\to 6}{\lim }\left(\frac{x^2-36}{x-6}\right)=12$

Explanation of Solution

Given:

The $\underset{x\to 6}{\lim }\left(\frac{x^2-36}{x-6}\right)$

Concept Used:

The limit of a function $f\left(x\right)$ at $x=c$ exists if the limit from right side of $c$ is equal to the limit from left side of $c$

  $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underset{x\to c^{+}}{\lim }f\left(x\right)$

Calculation:

For the $\underset{x\to 6}{\lim }\left(\frac{x^2-36}{x-6}\right)$

  $\begin{array}{c}\underset{x\to 6}{\lim }\left(\frac{x^2-36}{x-6}\right)=\underset{x\to 6}{\lim }\left(\frac{x^2-6^2}{x-6}\right)36=6^2\\ =\underset{x\to 6}{\lim }\left(\frac{\left(x+6\right)\left(x-6\right)}{x-6}\right)a^2-b^2=\left(a+b\right)\left(a-b\right)\\ =\underset{x\to 6}{\lim }\left(x+6\right)\text{ Divide out common factor}\\ =6+6\text{ Substituting the limit}\end{array}$

  $\text{=12}$

Therefore, $\underset{x\to 6}{\lim }\left(\frac{x^2-36}{x-6}\right)=12$

Using graphing utility, the graph of the function $f\left(x\right)=\frac{x^2-36}{x-6}$ is

  

Since, as $x$ tends to $6$ , the graph of the function $f\left(x\right)=\frac{x^2-36}{x-6}$ tends to $12$ .

Therefore, the graph of the function $f\left(x\right)=\frac{x^2-36}{x-6}$ verifies the result.

Conclusion:

The $\underset{x\to 6}{\lim }\left(\frac{x^2-36}{x-6}\right)=12$