Calculus Examples

$\frac{lim_{x \to - 2} x + 2}{x^{3} + 8}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{lim_{x \to - 2} x + lim_{x \to - 2} 2}{x^{3} + 8}$

Step 1.2

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

$\frac{lim_{x \to - 2} x + 2}{x^{3} + 8}$

Step 2

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

$\frac{- 2 + 2}{x^{3} + 8}$

Step 3

Simplify the answer.

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Step 3.1

Add $- 2$ and $2$ .

$\frac{0}{x^{3} + 8}$

Step 3.2

Simplify the denominator.

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Step 3.2.1

Rewrite $8$ as $2^{3}$ .

$\frac{0}{x^{3} + 2^{3}}$

Step 3.2.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 2$ .

$\frac{0}{(x + 2) (x^{2} - x \cdot 2 + 2^{2})}$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Multiply $2$ by $- 1$ .

$\frac{0}{(x + 2) (x^{2} - 2 x + 2^{2})}$

Step 3.2.3.2

Raise $2$ to the power of $2$ .

$\frac{0}{(x + 2) (x^{2} - 2 x + 4)}$

Step 3.3

Divide $0$ by $(x + 2) (x^{2} - 2 x + 4)$ .

$0$