Calculus Examples

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

Step 1

Apply L'Hospital's rule.

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Step 1.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1.1

Take the limit of the numerator and the limit of the denominator.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Add $- 2$ and $2$ .

$\frac{0}{lim_{x \to - 2} x^{3} + 8}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{lim_{x \to - 2} x^{3} + lim_{x \to - 2} 8}$

Step 1.1.3.1.2

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to - 2} x)}^{3} + lim_{x \to - 2} 8}$

Step 1.1.3.1.3

Evaluate the limit of $8$ which is constant as $x$ approaches $- 2$ .

$\frac{0}{{(lim_{x \to - 2} x)}^{3} + 8}$

Step 1.1.3.2

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

$\frac{0}{{(- 2)}^{3} + 8}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Raise $- 2$ to the power of $3$ .

$\frac{0}{- 8 + 8}$

Step 1.1.3.3.2

Add $- 8$ and $8$ .

$\frac{0}{0}$

Step 1.1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to - 2} \frac{x + 2}{x^{3} + 8} = lim_{x \to - 2} \frac{\frac{d}{d x} [x + 2]}{\frac{d}{d x} [x^{3} + 8]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $x + 2$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [2]$ .

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

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$lim_{x \to - 2} \frac{1 + \frac{d}{d x} [2]}{\frac{d}{d x} [x^{3} + 8]}$

Step 1.3.4

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{1 + 0}{\frac{d}{d x} [x^{3} + 8]}$

Step 1.3.5

Add $1$ and $0$ .

$lim_{x \to - 2} \frac{1}{\frac{d}{d x} [x^{3} + 8]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{3} + 8$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8]$ .

$lim_{x \to - 2} \frac{1}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [8]}$

Step 1.3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$lim_{x \to - 2} \frac{1}{3 x^{2} + \frac{d}{d x} [8]}$

Step 1.3.8

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$lim_{x \to - 2} \frac{1}{3 x^{2} + 0}$

Step 1.3.9

Add $3 x^{2}$ and $0$ .

$lim_{x \to - 2} \frac{1}{3 x^{2}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to - 2} \frac{1}{x^{2}}$

Step 2.2

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

$\frac{1}{3} \cdot \frac{lim_{x \to - 2} 1}{lim_{x \to - 2} x^{2}}$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $- 2$ .

$\frac{1}{3} \cdot \frac{1}{lim_{x \to - 2} x^{2}}$

Step 2.4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{1}{3} \cdot \frac{1}{{(lim_{x \to - 2} x)}^{2}}$

Step 3

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

$\frac{1}{3} \cdot \frac{1}{{(- 2)}^{2}}$

Step 4

Simplify the answer.

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Step 4.1

Combine.

$\frac{1 \cdot 1}{3 {(- 2)}^{2}}$

Step 4.2

Multiply $1$ by $1$ .

$\frac{1}{3 {(- 2)}^{2}}$

Step 4.3

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

$\frac{1}{3 \cdot 4}$

Step 4.4

Multiply $3$ by $4$ .

$\frac{1}{12}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{12}$

Decimal Form:

$0.08\overline{3}$