Calculus Examples

$lim_{x \to 3} \frac{x^{2} + x - 12}{| x - 3 |}$

Step 1

Consider the left sided limit.

$lim_{x \to 3^{-}} \frac{x^{2} + x - 12}{| x - 3 |}$

Step 2

Make a table to show the behavior of the function $\frac{x^{2} + x - 12}{| x - 3 |}$ as $x$ approaches $3$ from the left.

$\begin{array}{c} x & \frac{x^{2} + x - 12}{| x - 3 |} \\ 2.9 & - 6.9 \\ 2.99 & - 6.99 \\ 2.999 & - 6.999 \\ 2.9999 & - 6.9999 \\ 2.99999 & - 6.99999 \end{array}$

Step 3

As the $x$ values approach $3$ , the function values approach $- 7$ . Thus, the limit of $\frac{x^{2} + x - 12}{| x - 3 |}$ as $x$ approaches $3$ from the left is $- 7$ .

$- 7$

Step 4

Consider the right sided limit.

$lim_{x \to 3^{+}} \frac{x^{2} + x - 12}{| x - 3 |}$

Step 5

Make a table to show the behavior of the function $\frac{x^{2} + x - 12}{| x - 3 |}$ as $x$ approaches $3$ from the right.

$\begin{array}{c} x & \frac{x^{2} + x - 12}{| x - 3 |} \\ 3.1 & 7.1 \\ 3.01 & 7.01 \\ 3.001 & 7.001 \\ 3.0001 & 7.0001 \\ 3.00001 & 7.00001 \end{array}$

Step 6

As the $x$ values approach $3$ , the function values approach $7$ . Thus, the limit of $\frac{x^{2} + x - 12}{| x - 3 |}$ as $x$ approaches $3$ from the right is $7$ .

$7$

Step 7

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$