Chapter 14, Problem 41RE

To determine

To find:

The value of function $f\left(x\right)=\underset{x\to 2^{-}}{\lim }\frac{x^2-4}{x^3-8}$

Answer to Problem 41RE

  $\underset{x\to 2^{-}}{\lim }f\left(x\right)=\frac{1}{3}$

Explanation of Solution

Given:

The function $f$ is

  $f\left(x\right)=\underset{x\to 2^{-}}{\lim }\frac{x^2-4}{x^3-8}$

Concept used:

Let $f\left(x\right)$ be a function defined on an interval that contains $x=a$ , except possibly at $x=a$

Then

  $\underset{x\to a}{\lim }f\left(x\right)=L$

Calculation:

Let $f\left(x\right)$ be a function defined on an interval that contains $x=a$ , except possibly at $x=a$

  $\underset{x\to a}{\lim }f\left(x\right)=L$

If for every number $\in >0$ there is some number $\delta >0$ such that

  $\left|f\left(x\right)-L\right|<\in \text{when}\text{}\text{ever}0<\left|x-a\right|<\delta$

Other definition of limit

Let $f\left(x\right)$ be a function defined on an interval that contains $x=a$

  $\begin{array}{l}\underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }\frac{x^2-4}{x^3-8}\\ factor\\ x^3-8=\left(x-2\right)\left(x^2+2x+4\right)\\ x^2-4=\left(x-2\right)\left(x+2\right)\\ \underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }\frac{\left(x-2\right)\left(x+2\right)}{\left(x-2\right)\left(x^2+2x+4\right)}\\ \underset{x\to 2^{-}}{\lim }f\left(x\right)=\underset{x\to 2^{-}}{\lim }\frac{\left(x+2\right)}{\left(x^2+2x+4\right)}\\ \text{apply}\text{by}\text{limit}\\ \underset{x\to 2^{-}}{\lim }f\left(x\right)=\frac{\left(2+2\right)}{\left(2^2+2\left(2\right)+4\right)}\\ \underset{x\to 2^{-}}{\lim }f\left(x\right)=\frac{4}{\left(4+4+4\right)}\\ \underset{x\to 2^{-}}{\lim }f\left(x\right)=\frac{1}{3}\end{array}$