Chapter 11.2, Problem 68E

To determine

To calculate:

The value of $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$ for the following function:

  $f\left(x\right)=5+7x$

Answer to Problem 68E

The value of $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$ for the function $f\left(x\right)=5+7x$ is $7$ .

Explanation of Solution

Given information:

  $\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

  $f\left(x\right)=5+7x$

Formula used:

The slope of graph function is given by the following expression:

  $m=\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}$

Calculation:

The given function is:

  $f\left(x\right)=5+7x$

The function $f\left(x\right)=5+7x$ is given by the following expression:

  $\begin{array}{l}\underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{5+7\left(x+h\right)-5-7x}{h}\\ \underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{5+7\left(x+h\right)-5-7x}{h}\\ \underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{5+7x+7h-5-7x}{h}\\ \underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }\frac{7h}{h}\\ \underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=\underset{h\to 0}{\lim }7\left[\because h=0\right]\\ \underset{h\to 0}{\lim }\frac{f\left(x+h\right)-f\left(x\right)}{h}=7\end{array}$