Chapter 14, Problem 2CT

To determine

To calculate: The $\underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}$.

Answer to Problem 2CT

Solution:

The $\underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}$ is $\frac{1}{3}$.

Explanation of Solution

Given information:

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

Formula used:

$\underset{x\to c}{\lim }\left(\text{A}\right)=\text{A}$, where $\text{A}$ is constant

Calculation:

To determine $\underset{x\to 2^{+}}{\lim }f\left(x\right)$ means find the value of $f$ when $x$ close to $2$ but greater than $2$.

Thus, $|x-2|=x-2$, so function becomes $f\left(x\right)=\frac{x-2}{3x-6}$.

Thus, $\underset{x\to 2^{+}}{\lim }f\left(x\right)=\underset{x\to 2^{+}}{\lim }\frac{x-2}{3x-6}$

By factoring the denominator,

$=\underset{x\to 2^{+}}{\lim }\frac{x-2}{3\left(x-2\right)}$

By simplifying,

$=\underset{x\to 2^{+}}{\lim }\frac{1}{3}$ By using the property $\underset{x\to c}{\lim }\left(\text{A}\right)=\text{A}$,

$=\frac{1}{3}$ Therefore, the $\underset{x\to 2^{+}}{\lim }\frac{\left|x-2\right|}{3x-6}$ is $\frac{1}{3}$.