Calculus Examples

$lim_{x \to - 2} \frac{F (x^{2} - 4)}{| x + 2 |}$

Step 1

Set up the limit as a left-sided limit.

$lim_{x \to {(- 2)}^{-}} \frac{F (x^{2} - 4)}{| x + 2 |}$

Step 2

Evaluate the left-sided limit.

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Step 2.1

Move the term $F$ outside of the limit because it is constant with respect to $x$ .

$F lim_{x \to {(- 2)}^{-}} \frac{x^{2} - 4}{| x + 2 |}$

Step 2.2

Make a table to show the behavior of the function $\frac{x^{2} - 4}{| x + 2 |}$ as $x$ approaches $- 2$ from the left.

$\begin{array}{c} x & \frac{x^{2} - 4}{| x + 2 |} \\ - 2.1 & 4.1 \\ - 2.01 & 4.01 \\ - 2.001 & 4.001 \\ - 2.0001 & 4.0001 \\ - 2.00001 & 4.00001 \end{array}$

Step 2.3

As the $x$ values approach $- 2$ , the function values approach $4$ . Thus, the limit of $\frac{x^{2} - 4}{| x + 2 |}$ as $x$ approaches $- 2$ from the left is $4$ .

$F \cdot 4$

Step 2.4

Move $4$ to the left of $F$ .

$4 F$

Step 3

Set up the limit as a right-sided limit.

$lim_{x \to {(- 2)}^{+}} \frac{F (x^{2} - 4)}{| x + 2 |}$

Step 4

Evaluate the right-sided limit.

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Step 4.1

Move the term $F$ outside of the limit because it is constant with respect to $x$ .

$F lim_{x \to {(- 2)}^{+}} \frac{x^{2} - 4}{| x + 2 |}$

Step 4.2

Make a table to show the behavior of the function $\frac{x^{2} - 4}{| x + 2 |}$ as $x$ approaches $- 2$ from the right.

$\begin{array}{c} x & \frac{x^{2} - 4}{| x + 2 |} \\ - 1.9 & - 3.9 \\ - 1.99 & - 3.99 \\ - 1.999 & - 3.999 \\ - 1.9999 & - 3.9999 \\ - 1.99999 & - 3.99999 \end{array}$

Step 4.3

As the $x$ values approach $- 2$ , the function values approach $- 4$ . Thus, the limit of $\frac{x^{2} - 4}{| x + 2 |}$ as $x$ approaches $- 2$ from the right is $- 4$ .

$F \cdot - 4$

Step 4.4

Move $- 4$ to the left of $F$ .

$- 4 F$

Step 5

Since the left-sided limit is not equal to the right-sided limit, the limit does not exist.

$Does not exist$