Chapter 3, Problem 11RQ

To determine

The given statement is ture or false.

Answer to Problem 11RQ

This example is true .

Explanation of Solution

Given:

The function $g$ is

  $g\left(x\right)=x^5\text{and}\underset{x\to 2}{\lim }\frac{g\left(x\right)-g\left(2\right)}{x-2}=80$

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:

  $g\left(x\right)=x^5\mathrm{..................}\left(1\right)$

In general

  $\begin{array}{l}{a}^5-b^5=\left(a-b\right)\left(a^4+a^{3}b+a^2{b}^2+a{b}^3+b^4\right)\\ x^5-32=\left(x-2\right)\left(x^4+2x^3+2^2{x}^2+2^{3}x+2^4\right)\\ x^5-32=\left(x-2\right)\left(x^4+2x^3+4x^2+8x+16\right)\end{array}$

  $\begin{array}{l}\underset{x\to 2}{\lim }\frac{g\left(x\right)-g\left(2\right)}{x-2}\\ \underset{x\to 2}{\lim }\frac{x^5-32}{x-2}\\ \underset{x\to 2}{\lim }\frac{\left(x-2\right)\left(x^4+2x^3+4x^2+8x+16\right)}{x-2}\\ \underset{x\to 2}{\lim }\left(x^4+2x^3+4x^2+8x+16\right)\end{array}$

Applying from the limit

  $\begin{array}{l}\left(2^4+2\cdot 2^3+4\cdot 2^2+8\cdot 2+16\right)\\ =80\end{array}$

This example is true