Chapter 12.2, Problem 34E

a.

To determine

To find: To approximate the limit by using a graphing utility to graph the function.

Answer to Problem 34E

The approximate limit by using a graphing utility to graph the given function is $0.35$ .

Explanation of Solution

Given:$\underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}$

  

The above graph is the graphical representation of the given function with the limits $\underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}$ . From the graph we observe the limit $0.35$ .

Thus we can find the limit using graphing utility.

b.

To determine

To find: To numerically approximate the limit by using the table feature of the graphing utility to create a table.

Answer to Problem 34E

The numerically approximate limit using a table from the graph is $0.35$ .

Explanation of Solution

Given:$\underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}$

From the graph we can create a table,

x	f(x)
-0.5	0.378937
-0.4	0.373256
-0.3	0.36791
-0.2	0.362864
-0.1	0.358087
0	Indeterminate
0.1	0.349241

Therefore we can conclude that numerically the limit of the given function does is $0.35$ .

c.

To determine

To find: To algebraically evaluate the limit by using the appropriate techniques.

Answer to Problem 34E

By evaluating the given limit algebraically the limit is $0.35$ .

Explanation of Solution

Given:$\underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}$

Initially factor the numerator and denominator,

  $\begin{array}{l}\underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}=\underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}\cdot \frac{\sqrt{x+2}+\sqrt{2}}{\sqrt{x+2}+\sqrt{2}}\\ \underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}=\underset{x\to 0^{-}}{\lim }\frac{x+2-2}{x\left(\sqrt{x+2}+\sqrt{2}\right)}\\ \underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}=\underset{x\to 0^{-}}{\lim }\frac{1}{\sqrt{x+2}+\sqrt{2}},\text{ by using direct substitution}\\ \underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}=\frac{1}{2\sqrt{2}}\\ \underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}=0.354\end{array}$

Thus we can approximate the limit algebraically.