Question

In Problems 1– 6, find each limit

$\underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}$

Short answer

The value of the limit is$\frac{1}{3}$

Step-by-step solution

Step 1. Given data

Given limit is $\underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}$

We have to find the value of the limit.

Step 2. Rewriting the limit

Since the given limit is $x\to 2^{+}$, means $x-2$is positive, we can remove the modulus in the numerator.

$$\begin{gathered} \underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}=\underset{x\to 2^{+}}{\lim }\frac{x-2}{3(x-2)} \\ \Rightarrow \underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}=\underset{x\to 2^{+}}{\lim }\frac{1}{3} \\ \Rightarrow \underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}=\frac{1}{3} \end{gathered}$$