Chapter 12.3, Problem 19E

To determine

To find the slope of the graph of the given function at the given point by using limit process.

Answer to Problem 19E

  $m=-1$

Explanation of Solution

Given:

  $g\left(x\right)=\frac{4}{x},\left(2,2\right)$

Concept used:

The slope of a graph of a function $f$ at a specific point $\left(c,f\left(c\right)\right)$ is given by,

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

Calculation:

Given,

  $g\left(x\right)=\frac{4}{x},\left(2,2\right)$

Now slope of the function $g$ at $\left(2,2\right)$ is,

  $\begin{array}{l}m=\underset{h\to 0}{\lim }\frac{g\left(2+h\right)-g\left(2\right)}{h}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{\frac{4}{2+h}-\frac{4}{2}}{h}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{\frac{4}{2+h}-2}{h}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{\frac{4-2\left(2+h\right)}{2+h}}{h}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{-2h}{h\left(2+h\right)}\\ \Rightarrow m=\underset{h\to 0}{\lim }\frac{-2}{2+h}\\ \Rightarrow m=\frac{-2}{2+0}\\ \Rightarrow m=-1\end{array}$