Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 1.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $- 2$ .

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

Step 1.3

Evaluate the limit of $3$ which is constant as $x$ approaches $- 2$ .

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

Step 2

Evaluate the limit of $x$ by plugging in $- 2$ for $x$ .

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

Step 3

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

$\begin{array}{c} x & \frac{| x + 2 |}{x + 2} \\ - 2.1 & - 1 \\ - 2.01 & - 1 \\ - 2.001 & - 1 \end{array}$

Step 4

As the $x$ values approach $- 2$ , the function values approach $- 1$ . Thus, the limit of $\frac{| x + 2 |}{x + 2}$ as $x$ approaches $- 2$ from the left is $- 1$ .

$(- 2 + 3) \cdot - 1$

Step 5

Simplify the answer.

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Step 5.1

Add $- 2$ and $3$ .

$1 \cdot - 1$

Step 5.2

Multiply $- 1$ by $1$ .

$- 1$