Chapter 11.2, Problem 24E

To determine

To find: the value of the limit.

Answer to Problem 24E

The value of the limit is $\frac{-1}{2\sqrt{7}}$

Explanation of Solution

Given:

  $\underset{z\to 0}{\lim }\frac{\sqrt{7-z}-\sqrt{7}}{z}$

Calculation:

  $\begin{array}{l}\underset{z\to 0}{\lim }\frac{\sqrt{7-z}-\sqrt{5}}{z}=\underset{z\to 0}{\lim }\frac{\sqrt{7-z}-\sqrt{7}}{z}\cdot \frac{\sqrt{7-z}+\sqrt{7}}{\sqrt{7-z}+\sqrt{7}}\\ =\underset{z\to 0}{\lim }\frac{7-z-7}{z\left(\sqrt{7-z}+\sqrt{7}\right)}\\ =\underset{z\to 0}{\lim }\frac{-z}{z\left(\sqrt{7-z}+\sqrt{7}\right)}\\ =\underset{z\to 0}{\lim }\frac{-1}{\left(\sqrt{7-z}+\sqrt{7}\right)}\\ =\frac{-1}{2\sqrt{7}}\left(By\text{ direct substitution}\right)\end{array}$

See this from the graph of the function:

  

Conclusion:

The value of the limit is $\frac{-1}{2\sqrt{7}}$