Calculus Examples

$lim_{x \to 3} (x^{2} - 9) \frac{x - 3}{| x - 3 |}$

Step 1

Multiply $x^{2} - 9$ by $\frac{x - 3}{| x - 3 |}$ .

$lim_{x \to 3} \frac{(x^{2} - 9) (x - 3)}{| x - 3 |}$

Step 2

Consider the left sided limit.

$lim_{x \to 3^{-}} \frac{(x^{2} - 9) (x - 3)}{| x - 3 |}$

Step 3

Make a table to show the behavior of the function $\frac{(x^{2} - 9) (x - 3)}{| x - 3 |}$ as $x$ approaches $3$ from the left.

$\begin{array}{c} x & \frac{(x^{2} - 9) (x - 3)}{| x - 3 |} \\ 2.9 & 0.59 \\ 2.99 & 0.0599 \\ 2.999 & 0.005999 \\ 2.9999 & 0.0005\overline{9} \\ 2.99999 & 0.00005\overline{9} \\ 2.999999 & 0.000005\overline{9} \end{array}$

Step 4

As the $x$ values approach $3$ , the function values approach $0$ . Thus, the limit of $\frac{(x^{2} - 9) (x - 3)}{| x - 3 |}$ as $x$ approaches $3$ from the left is $0$ .

$0$

Step 5

Consider the right sided limit.

$lim_{x \to 3^{+}} \frac{(x^{2} - 9) (x - 3)}{| x - 3 |}$

Step 6

Make a table to show the behavior of the function $\frac{(x^{2} - 9) (x - 3)}{| x - 3 |}$ as $x$ approaches $3$ from the right.

$\begin{array}{c} x & \frac{(x^{2} - 9) (x - 3)}{| x - 3 |} \\ 3.1 & 0.61 \\ 3.01 & 0.0601 \\ 3.001 & 0.006001 \\ 3.0001 & 0.00060001 \\ 3.00001 & 0.00006 \\ 3.000001 & 0.000006 \end{array}$

Step 7

As the $x$ values approach $3$ , the function values approach $0$ . Thus, the limit of $\frac{(x^{2} - 9) (x - 3)}{| x - 3 |}$ as $x$ approaches $3$ from the right is $0$ .

$0$