Chapter 11.2, Problem 60E

a.

To determine

To graphically approximate the limit by using a graphing utility to graph the function $\frac{\sqrt{x+2}-\sqrt{2}}{x}$ .

Answer to Problem 60E

  $\frac{1}{2\sqrt{2}}\text{or 0}\text{.353}$

Explanation of Solution

Given:

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

Graph:

  

Graph has been plotted and its value at $x=0$ is $0.353$ .

b.

To determine

To numerically approximate the limit using graphing utility of the function $\frac{\sqrt{x+2}-\sqrt{2}}{x}$ .

Explanation of Solution

Calculation:

Make a table for the values of function obtained after substituting corresponding values of x as shown below.

  $y=\frac{\sqrt{x+2}-\sqrt{2}}{x}$

x	-0.9	-0.99	-0.999	0	0.1	0.01	0.001
y	1.266	-1.4	-0.585	0.353	0.35	0.3	0

its value at $x=0$ is $0.353$ .

c.

To determine

To evaluate the limit of expression $\frac{\sqrt{x+2}-\sqrt{2}}{x}$ .

Answer to Problem 60E

  $\frac{1}{2\sqrt{2}}\text{or 0}\text{.353}$

Explanation of Solution

Given:

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

Calculation:

First rationalise the function and then substitute value of x as 0 in the simplified expression.

  $\begin{array}{l}\underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}\\ \underset{x\to 0^{-}}{\lim }\frac{\sqrt{x+2}-\sqrt{2}}{x}\times \frac{\sqrt{x+2}+\sqrt{2}}{\sqrt{x+2}+\sqrt{2}}\\ =\underset{x\to 0^{-}}{\lim }\frac{{(\sqrt{x+2})}^2-{(\sqrt{2})}^2}{(x)(\sqrt{x+2}+\sqrt{2})}\\ =\underset{x\to 0^{-}}{\lim }\frac{(x+2-2)}{(x)(\sqrt{x+2}+\sqrt{2})}\\ =\underset{x\to 0^{-}}{\lim }\frac{1}{(\sqrt{x+2}+\sqrt{2})}\\ =\frac{1}{(\sqrt{0+2}+\sqrt{2})}\\ =\frac{1}{2\sqrt{2}}\text{or 0}\text{.353}\end{array}$