Chapter 12.1, Problem 13E

To determine

Graph a table of values for the function $\underset{x\to 0}{\lim }\frac{\sqrt{x+5}-\sqrt{5}}{x}$ and estimate the limit numerically.

Answer to Problem 13E

  $\frac{1}{2\sqrt{5}}$

Explanation of Solution

Given:

  $\underset{x\to 0}{\lim }\frac{\sqrt{x+5}-\sqrt{5}}{x}$

Concept Used:

When solving for limits, substitute values that are close to the value of $x$ to obtain a define value of $f(x)$ . If the substituted value is equal to the value of $x$ , the value of $f(x)$ becomes undefined. This means that as $x$ approaches to a certain value, the limit still exists.

Calculation for graph:

Consider $f\left(x\right)=\underset{x\to 0}{\lim }\frac{\sqrt{x+5}-\sqrt{5}}{x}$

Values of x	Values of f ( x )
0.001	0.2236
0.01	0.2235
0.1	0.2225

By taking different values of x , the graph can be plotted.

The table shows that as $x$ approaches to 0, the value of $f(x)$ approaches to $\frac{1}{2\sqrt{5}}$ or 0.2236.

Graph:

  

Interpretation:

The graph shows that as $x$ approaches 0, the value of $f\left(x\right)$ approaches $\frac{1}{2\sqrt{5}}$ or 0.2236.

Calculation:

Estimating the limit numerically,

  $\begin{array}{l}\Rightarrow \underset{x\to 0}{\lim }\left(\frac{\sqrt{x+5}-\sqrt{5}}{x}\right)\left(\frac{\sqrt{x+5}+\sqrt{5}}{\sqrt{x+5}+\sqrt{5}}\right)\\ \Rightarrow \underset{x\to 0}{\lim }\frac{{\left(\sqrt{x+5}\right)}^2-\left(\sqrt{5}\right)\left(\sqrt{x+5}\right)+\left(\sqrt{5}\right)\left(\sqrt{x+5}\right)+{\left(\sqrt{5}\right)}^2}{x\left(\sqrt{x+5}+\sqrt{5}\right)}\\ \Rightarrow \underset{x\to 0}{\lim }\frac{{\left(\sqrt{x+5}\right)}^2-{\left(\sqrt{5}\right)}^2}{x\left(\sqrt{x+5}+\sqrt{5}\right)}\\ \Rightarrow \underset{x\to 0}{\lim }\frac{\left(x+5\right)-5}{x\left(\sqrt{x+5}+\sqrt{5}\right)}\\ \Rightarrow \underset{x\to 0}{\lim }\frac{x}{x\left(\sqrt{x+5}+\sqrt{5}\right)}\\ \Rightarrow \underset{x\to 0}{\lim }\frac{1}{\sqrt{x+5}+\sqrt{5}}\\ \Rightarrow \frac{1}{\sqrt{0+5}+\sqrt{5}}\\ \Rightarrow \frac{1}{\sqrt{5}+\sqrt{5}}\\ \Rightarrow \frac{1}{2\sqrt{5}}\end{array}$

Conclusion:

Therefore, when $x$ approaches 0, the value of $f\left(x\right)$ approaches $\frac{1}{2\sqrt{5}}$ or 0.2236.