Calculus Examples

$lim_{x \to 16} \frac{\sqrt[4]{x^{7}} - 128}{x - 16}$

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.

$\frac{lim_{x \to 16} \sqrt[4]{x^{7}} - 128}{lim_{x \to 16} x - 16}$

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 $16$ .

$\frac{lim_{x \to 16} \sqrt[4]{x^{7}} - lim_{x \to 16} 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.1.2

Move the limit under the radical sign.

$\frac{\sqrt[4]{lim_{x \to 16} x^{7}} - lim_{x \to 16} 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.1.3

Move the exponent $7$ from $x^{7}$ outside the limit using the Limits Power Rule.

$\frac{\sqrt[4]{{(lim_{x \to 16} x)}^{7}} - lim_{x \to 16} 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.1.4

Evaluate the limit of $128$ which is constant as $x$ approaches $16$ .

$\frac{\sqrt[4]{{(lim_{x \to 16} x)}^{7}} - 1 \cdot 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $16$ for $x$ .

$\frac{\sqrt[4]{16^{7}} - 1 \cdot 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Raise $16$ to the power of $7$ .

$\frac{\sqrt[4]{268435456} - 1 \cdot 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.3.1.2

Rewrite $268435456$ as $128^{4}$ .

$\frac{\sqrt[4]{128^{4}} - 1 \cdot 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.3.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\frac{128 - 1 \cdot 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.3.1.4

Multiply $- 1$ by $128$ .

$\frac{128 - 128}{lim_{x \to 16} x - 16}$

Step 1.1.2.3.2

Subtract $128$ from $128$ .

$\frac{0}{lim_{x \to 16} x - 16}$

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 $16$ .

$\frac{0}{lim_{x \to 16} x - lim_{x \to 16} 16}$

Step 1.1.3.1.2

Evaluate the limit of $16$ which is constant as $x$ approaches $16$ .

$\frac{0}{lim_{x \to 16} x - 1 \cdot 16}$

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $16$ for $x$ .

$\frac{0}{16 - 1 \cdot 16}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $16$ .

$\frac{0}{16 - 16}$

Step 1.1.3.3.2

Subtract $16$ from $16$ .

$\frac{0}{0}$

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.

$lim_{x \to 16} \frac{\sqrt[4]{x^{7}} - 128}{x - 16} = lim_{x \to 16} \frac{\frac{d}{d x} [\sqrt[4]{x^{7}} - 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 16} \frac{\frac{d}{d x} [\sqrt[4]{x^{7}} - 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt[4]{x^{7}} - 128$ with respect to $x$ is $\frac{d}{d x} [\sqrt[4]{x^{7}}] + \frac{d}{d x} [- 128]$ .

$lim_{x \to 16} \frac{\frac{d}{d x} [\sqrt[4]{x^{7}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt[4]{x^{7}}]$ .

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{7}}$ as $x^{\frac{7}{4}}$ .

$lim_{x \to 16} \frac{\frac{d}{d x} [x^{\frac{7}{4}}] + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{4}$ .

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{7}{4} - 1} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{7}{4} - 1 \cdot \frac{4}{4}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.3.4

Combine $- 1$ and $\frac{4}{4}$ .

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{7}{4} + \frac{- 1 \cdot 4}{4}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.3.5

Combine the numerators over the common denominator.

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{7 - 1 \cdot 4}{4}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $4$ .

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{7 - 4}{4}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.3.6.2

Subtract $4$ from $7$ .

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{3}{4}} + \frac{d}{d x} [- 128]}{\frac{d}{d x} [x - 16]}$

Step 1.3.4

Since $- 128$ is constant with respect to $x$ , the derivative of $- 128$ with respect to $x$ is $0$ .

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{3}{4}} + 0}{\frac{d}{d x} [x - 16]}$

Step 1.3.5

Simplify.

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Step 1.3.5.1

Add $\frac{7}{4} x^{\frac{3}{4}}$ and $0$ .

$lim_{x \to 16} \frac{\frac{7}{4} x^{\frac{3}{4}}}{\frac{d}{d x} [x - 16]}$

Step 1.3.5.2

Combine $\frac{7}{4}$ and $x^{\frac{3}{4}}$ .

$lim_{x \to 16} \frac{\frac{7 x^{\frac{3}{4}}}{4}}{\frac{d}{d x} [x - 16]}$

Step 1.3.6

By the Sum Rule, the derivative of $x - 16$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 16]$ .

$lim_{x \to 16} \frac{\frac{7 x^{\frac{3}{4}}}{4}}{\frac{d}{d x} [x] + \frac{d}{d x} [- 16]}$

Step 1.3.7

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

$lim_{x \to 16} \frac{\frac{7 x^{\frac{3}{4}}}{4}}{1 + \frac{d}{d x} [- 16]}$

Step 1.3.8

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16$ with respect to $x$ is $0$ .

$lim_{x \to 16} \frac{\frac{7 x^{\frac{3}{4}}}{4}}{1 + 0}$

Step 1.3.9

Add $1$ and $0$ .

$lim_{x \to 16} \frac{\frac{7 x^{\frac{3}{4}}}{4}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 16} \frac{7 x^{\frac{3}{4}}}{4} \cdot 1$

Step 1.5

Multiply $\frac{7 x^{\frac{3}{4}}}{4}$ by $1$ .

$lim_{x \to 16} \frac{7 x^{\frac{3}{4}}}{4}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{7}{4}$ outside of the limit because it is constant with respect to $x$ .

$\frac{7}{4} lim_{x \to 16} x^{\frac{3}{4}}$

Step 2.2

Move the exponent $\frac{3}{4}$ from $x^{\frac{3}{4}}$ outside the limit using the Limits Power Rule.

$\frac{7}{4} {(lim_{x \to 16} x)}^{\frac{3}{4}}$

Step 3

Evaluate the limit of $x$ by plugging in $16$ for $x$ .

$\frac{7}{4} \cdot 16^{\frac{3}{4}}$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $16$ as $2^{4}$ .

$\frac{7}{4} \cdot {(2^{4})}^{\frac{3}{4}}$

Step 4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{7}{4} \cdot 2^{4 (\frac{3}{4})}$

Step 4.3

Cancel the common factor of $4$ .

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Step 4.3.1

Cancel the common factor.

$\frac{7}{4} \cdot 2^{4 (\frac{3}{4})}$

Step 4.3.2

Rewrite the expression.

$\frac{7}{4} \cdot 2^{3}$

Step 4.4

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

$\frac{7}{4} \cdot 8$

Step 4.5

Cancel the common factor of $4$ .

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Step 4.5.1

Factor $4$ out of $8$ .

$\frac{7}{4} \cdot (4 (2))$

Step 4.5.2

Cancel the common factor.

$\frac{7}{4} \cdot (4 \cdot 2)$

Step 4.5.3

Rewrite the expression.

$7 \cdot 2$

Step 4.6

Multiply $7$ by $2$ .

$14$