Calculus Examples

$lim_{x \to - 5} \frac{- 8 | x | + 40}{2 x + 10}$

Step 1

Cancel the common factor of $- 8 | x | + 40$ and $2 x + 10$ .

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Step 1.1

Factor $2$ out of $- 8 | x |$ .

$lim_{x \to - 5} \frac{2 (- 4 | x |) + 40}{2 x + 10}$

Step 1.2

Factor $2$ out of $40$ .

$lim_{x \to - 5} \frac{2 (- 4 | x |) + 2 \cdot 20}{2 x + 10}$

Step 1.3

Factor $2$ out of $2 (- 4 | x |) + 2 (20)$ .

$lim_{x \to - 5} \frac{2 (- 4 | x | + 20)}{2 x + 10}$

Step 1.4

Cancel the common factors.

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Step 1.4.1

Factor $2$ out of $2 x$ .

$lim_{x \to - 5} \frac{2 (- 4 | x | + 20)}{2 (x) + 10}$

Step 1.4.2

Factor $2$ out of $10$ .

$lim_{x \to - 5} \frac{2 (- 4 | x | + 20)}{2 x + 2 \cdot 5}$

Step 1.4.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

$lim_{x \to - 5} \frac{2 (- 4 | x | + 20)}{2 (x + 5)}$

Step 1.4.4

Cancel the common factor.

$lim_{x \to - 5} \frac{2 (- 4 | x | + 20)}{2 (x + 5)}$

Step 1.4.5

Rewrite the expression.

$lim_{x \to - 5} \frac{- 4 | x | + 20}{x + 5}$

Step 2

Consider the left sided limit.

$lim_{x \to {(- 5)}^{-}} \frac{- 4 | x | + 20}{x + 5}$

Step 3

Make a table to show the behavior of the function $\frac{- 4 | x | + 20}{x + 5}$ as $x$ approaches $- 5$ from the left.

$\begin{array}{c} x & \frac{- 4 | x | + 20}{x + 5} \\ - 5.1 & 4 \\ - 5.01 & 4 \\ - 5.001 & 4 \end{array}$

Step 4

As the $x$ values approach $- 5$ , the function values approach $4$ . Thus, the limit of $\frac{- 4 | x | + 20}{x + 5}$ as $x$ approaches $- 5$ from the left is $4$ .

$4$

Step 5

Consider the right sided limit.

$lim_{x \to {(- 5)}^{+}} \frac{- 4 | x | + 20}{x + 5}$

Step 6

Make a table to show the behavior of the function $\frac{- 4 | x | + 20}{x + 5}$ as $x$ approaches $- 5$ from the right.

$\begin{array}{c} x & \frac{- 4 | x | + 20}{x + 5} \\ - 4.9 & 4 \\ - 4.99 & 4 \\ - 4.999 & 4 \end{array}$

Step 7

As the $x$ values approach $- 5$ , the function values approach $4$ . Thus, the limit of $\frac{- 4 | x | + 20}{x + 5}$ as $x$ approaches $- 5$ from the right is $4$ .

$4$