Question

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

$\underset{x\to 3}{\lim }\frac{x^2-x-6}{x-3}$

Short answer

The value of limit of the given function is$5$.

Step-by-step solution

Step 1. Given information.

The function here given is,$\underset{x\to 3}{\lim }\frac{x^2-x-6}{x-3}$

Step 2. Simplification.

Given, $\underset{x\to 3}{\lim }\frac{x^2-x-6}{x-3}$

Using factorisation method we get,

$\begin{array}{c}\underset{x\to 3}{\lim }\frac{x^2-x-6}{x-3}\\ \underset{x\to 3}{\lim }\frac{x^2-3x+2x-6}{x-3}\\ \underset{x\to 3}{\lim }\frac{x\left(x-3\right)+2(x-3)}{x-3}\\ \underset{x\to 3}{\lim }\frac{\left(x-3\right)\left(x+2\right)}{\left(x-3\right)}\\ \underset{x\to 3}{\lim }(x+2)\end{array}$

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.

$x<3$	$f\left(x\right)$
$2.99$	$4.99$
$2.98$	$4.98$
$2.97$	$4.97$
$2.96$	$4.96$

$x>3$	$f\left(x\right)$
$3.01$	$5.01$
$3.02$	$5.02$
$3.03$	$5.03$
$3.04$	$5.04$

On the basis of the values in the two tables we make the guess that,$\underset{x\to 3}{\lim }\frac{x^2-x-6}{x-3}=5$

Step 5. Draw the graph.

Now, as a graphical verification use a graphing device.

Therefore, from the graph it is verified that$\underset{x\to 3}{\lim }\frac{x^2-x-6}{x-3}=5$