Question

Estimate the value of the limit by making a table of values. Check your work with a graph.

$\underset{x\to 5}{\lim }\frac{x^2-25}{x-5}$

Short answer

The value of limit of the given function is$10$.

Step-by-step solution

Step 1. Given information.

The limit of function here given is,

$\underset{x\to 5}{\lim }\frac{x^2-25}{x-5}$

Step 2. Formula used.

The algebraic formula will be used here to simplify. That is,$a^2-b^2=\left(a+b\right)\left(a-b\right)$

Step 3. Simplification.

Given,$\underset{x\to 5}{\lim }\frac{x^2-25}{x-5}$

Using the formula we get,

$\begin{array}{c}\underset{x\to 5}{\lim }\frac{x^2-5^2}{x-5}\\ \underset{x\to 5}{\lim }\frac{\left(x+5\right)\left(x-5\right)}{(x-5)}\\ \underset{x\to 5}{\lim }\left(x+5\right)\end{array}$

Step 4. Make the table.

The following table gives the value of $f\left(x\right)$considering, values of$x$that are close to $5$, but not equal to$5$

$x<5$	$f\left(x\right)$
$4.99$	$9.99$
$4.98$	$9.98$
$4.97$	$9.97$
$4.96$	$9.96$

$x>5$$5.01$	$f\left(x\right)$
$5.01$	$10.01$ $10.01$
$5.02$	$10.02$
$5.03$	$10.03$
$5.04$	$10.04$

On the basis of the values in the two tables we make the guess that,

$\underset{x\to 5}{\lim }\frac{x^2-25}{x-5}=10$

Step 5. Draw the graph.

Now, as a graphical verification use a graphing device.

Therefore, from the graph it is verified that$\underset{x\to 5}{\lim }\frac{x^2-25}{x-5}=10$