Chapter 12.1, Problem 26E

To determine

ToEvaluate:Use the graph given below to find the limit $\underset{x\to 2}{\lim }\left(\frac{3x^2-12}{x-2}\right)$ , if it exists. If the limit does not exist, explain why.

  

Answer to Problem 26E

The given limit $\underset{x\to 2}{\lim }\left(\frac{3x^2-12}{x-2}\right)$ exists.

Explanation of Solution

Given:

The graph of the function $\underset{x\to 2}{\lim }\left(\frac{3x^2-12}{x-2}\right)$

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:

Given the graph of the function $\underset{x\to 2}{\lim }\left(\frac{3x^2-12}{x-2}\right)$

  

Left hand limit

From the graph of $f\left(x\right)=\frac{3x^2-12}{x-2}$ as $x$ approaches $2$ from left hand side, the graph of the function $f\left(x\right)=\frac{3x^2-12}{x-2}$ approaches to $12$ .

  $\underset{x\to 2^{-}}{\lim }\frac{3x^2-12}{x-2}=12$

Right hand limit

From the graph of $f\left(x\right)=\frac{3x^2-12}{x-2}$ as $x$ approaches $2$ from right hand side, the graph of the function $f\left(x\right)=\frac{3x^2-12}{x-2}$ approaches to $12$ .

  $\underset{x\to 2^{+}}{\lim }\frac{3x^2-12}{x-2}=12$

Since, $\underset{x\to 2^{-}}{\lim }\frac{3x^2-12}{x-2}=12=\underset{x\to 2^{+}}{\lim }\frac{3x^2-12}{x-2}$

Therefore, the given limit $\underset{x\to 2}{\lim }\left(\frac{3x^2-12}{x-2}\right)$ exists.

Conclusion:

The given limit $\underset{x\to 2}{\lim }\left(\frac{3x^2-12}{x-2}\right)$ exists.