Question

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically.

$\underset{x\to 0}{\lim }\frac{\sqrt{x+9}-3}{x}$

Short answer

The value of limit of the given function is$\frac{1}{6}$.

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

$\underset{x\to 0}{\lim }\frac{\sqrt{x+9}-3}{x}$

Step 2. Simplify.

Rationalize the numerator for simplification,

$\begin{array}{c}\frac{\sqrt{x+9}-3}{x}:\frac{1}{\sqrt{x+9}+3}\\ =\underset{x\to 0}{\lim }\left(\frac{1}{\sqrt{x+9}+3}\right)\end{array}$

Now, plug in the value $x=0$

$\begin{array}{c}=\frac{1}{\sqrt{0+9}+3}\\ =\frac{1}{6}\end{array}$

Step 3. Make the table.

Now make a table,

Therefore, on the basis of the values in the table we make the guess that,

$\underset{x\to 0}{\lim }\frac{\sqrt{x+9}-3}{x}=0.16=\frac{1}{6}$

Step 4. Draw the graph.

Now, as a graphical verification use a graphing device.

From the graph it is verified that,

$\underset{x\to 0}{\lim }\frac{\sqrt{x+9}-3}{x}=\frac{1}{6}$