Calculus Examples

$lim_{x \to 4} \frac{x - 4}{x^{3} - 64}$

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.

$\frac{lim_{x \to 4} x - 4}{lim_{x \to 4} x^{3} - 64}$

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 $4$ .

$\frac{lim_{x \to 4} x - lim_{x \to 4} 4}{lim_{x \to 4} x^{3} - 64}$

Step 1.1.2.1.2

Evaluate the limit of $4$ which is constant as $x$ approaches $4$ .

$\frac{lim_{x \to 4} x - 1 \cdot 4}{lim_{x \to 4} x^{3} - 64}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{4 - 1 \cdot 4}{lim_{x \to 4} x^{3} - 64}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $4$ .

$\frac{4 - 4}{lim_{x \to 4} x^{3} - 64}$

Step 1.1.2.3.2

Subtract $4$ from $4$ .

$\frac{0}{lim_{x \to 4} x^{3} - 64}$

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 $4$ .

$\frac{0}{lim_{x \to 4} x^{3} - lim_{x \to 4} 64}$

Step 1.1.3.1.2

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

$\frac{0}{{(lim_{x \to 4} x)}^{3} - lim_{x \to 4} 64}$

Step 1.1.3.1.3

Evaluate the limit of $64$ which is constant as $x$ approaches $4$ .

$\frac{0}{{(lim_{x \to 4} x)}^{3} - 1 \cdot 64}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{0}{4^{3} - 1 \cdot 64}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

Raise $4$ to the power of $3$ .

$\frac{0}{64 - 1 \cdot 64}$

Step 1.1.3.3.1.2

Multiply $- 1$ by $64$ .

$\frac{0}{64 - 64}$

Step 1.1.3.3.2

Subtract $64$ from $64$ .

$\frac{0}{0}$

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.

$lim_{x \to 4} \frac{x - 4}{x^{3} - 64} = lim_{x \to 4} \frac{\frac{d}{d x} [x - 4]}{\frac{d}{d x} [x^{3} - 64]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 4} \frac{\frac{d}{d x} [x - 4]}{\frac{d}{d x} [x^{3} - 64]}$

Step 1.3.2

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$lim_{x \to 4} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- 4]}{\frac{d}{d x} [x^{3} - 64]}$

Step 1.3.3

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

$lim_{x \to 4} \frac{1 + \frac{d}{d x} [- 4]}{\frac{d}{d x} [x^{3} - 64]}$

Step 1.3.4

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{1 + 0}{\frac{d}{d x} [x^{3} - 64]}$

Step 1.3.5

Add $1$ and $0$ .

$lim_{x \to 4} \frac{1}{\frac{d}{d x} [x^{3} - 64]}$

Step 1.3.6

By the Sum Rule, the derivative of $x^{3} - 64$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 64]$ .

$lim_{x \to 4} \frac{1}{\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 64]}$

Step 1.3.7

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

$lim_{x \to 4} \frac{1}{3 x^{2} + \frac{d}{d x} [- 64]}$

Step 1.3.8

Since $- 64$ is constant with respect to $x$ , the derivative of $- 64$ with respect to $x$ is $0$ .

$lim_{x \to 4} \frac{1}{3 x^{2} + 0}$

Step 1.3.9

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

$lim_{x \to 4} \frac{1}{3 x^{2}}$

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 $x$ .

$\frac{1}{3} lim_{x \to 4} \frac{1}{x^{2}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $4$ .

$\frac{1}{3} \cdot \frac{lim_{x \to 4} 1}{lim_{x \to 4} x^{2}}$

Step 2.3

Evaluate the limit of $1$ which is constant as $x$ approaches $4$ .

$\frac{1}{3} \cdot \frac{1}{lim_{x \to 4} x^{2}}$

Step 2.4

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

$\frac{1}{3} \cdot \frac{1}{{(lim_{x \to 4} x)}^{2}}$

Step 3

Evaluate the limit of $x$ by plugging in $4$ for $x$ .

$\frac{1}{3} \cdot \frac{1}{4^{2}}$

Step 4

Simplify the answer.

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Step 4.1

Combine.

$\frac{1 \cdot 1}{3 \cdot 4^{2}}$

Step 4.2

Multiply $1$ by $1$ .

$\frac{1}{3 \cdot 4^{2}}$

Step 4.3

Raise $4$ to the power of $2$ .

$\frac{1}{3 \cdot 16}$

Step 4.4

Multiply $3$ by $16$ .

$\frac{1}{48}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{48}$

Decimal Form:

$0.0208\overline{3}$