Calculus Examples

$lim_{h \to 0} \frac{2 \sqrt[3]{8 + h} - 2 \sqrt[3]{8}}{h}$

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.

$\frac{lim_{h \to 0} 2 \sqrt[3]{8 + h} - 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

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 $h$ approaches $0$ .

$\frac{lim_{h \to 0} 2 \sqrt[3]{8 + h} - lim_{h \to 0} 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.1.2

Move the term $2$ outside of the limit because it is constant with respect to $h$ .

$\frac{2 lim_{h \to 0} \sqrt[3]{8 + h} - lim_{h \to 0} 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.1.3

Move the limit under the radical sign.

$\frac{2 \sqrt[3]{lim_{h \to 0} 8 + h} - lim_{h \to 0} 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.1.4

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{2 \sqrt[3]{lim_{h \to 0} 8 + lim_{h \to 0} h} - lim_{h \to 0} 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.1.5

Evaluate the limit of $8$ which is constant as $h$ approaches $0$ .

$\frac{2 \sqrt[3]{8 + lim_{h \to 0} h} - lim_{h \to 0} 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.1.6

Evaluate the limit of $2 \sqrt[3]{8}$ which is constant as $h$ approaches $0$ .

$\frac{2 \sqrt[3]{8 + lim_{h \to 0} h} - 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.2

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{2 \sqrt[3]{8 + 0} - 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

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

Add $8$ and $0$ .

$\frac{2 \sqrt[3]{8} - 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.3.1.2

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

$\frac{2 \sqrt[3]{2^{3}} - 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.3.1.3

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

$\frac{2 \cdot 2 - 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.3.1.4

Multiply $2$ by $2$ .

$\frac{4 - 2 \sqrt[3]{8}}{lim_{h \to 0} h}$

Step 1.1.2.3.1.5

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

$\frac{4 - 2 \sqrt[3]{2^{3}}}{lim_{h \to 0} h}$

Step 1.1.2.3.1.6

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

$\frac{4 - 2 \cdot 2}{lim_{h \to 0} h}$

Step 1.1.2.3.1.7

Multiply $- 2$ by $2$ .

$\frac{4 - 4}{lim_{h \to 0} h}$

Step 1.1.2.3.2

Subtract $4$ from $4$ .

$\frac{0}{lim_{h \to 0} h}$

Step 1.1.3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{0}{0}$

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.

$lim_{h \to 0} \frac{2 \sqrt[3]{8 + h} - 2 \sqrt[3]{8}}{h} = lim_{h \to 0} \frac{\frac{d}{d h} [2 \sqrt[3]{8 + h} - 2 \sqrt[3]{8}]}{\frac{d}{d h} [h]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{h \to 0} \frac{\frac{d}{d h} [2 \sqrt[3]{8 + h} - 2 \sqrt[3]{8}]}{\frac{d}{d h} [h]}$

Step 1.3.2

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

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

Step 1.3.3

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

$lim_{h \to 0} \frac{\frac{d}{d h} [2 \sqrt[3]{8 + h} - 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.4

By the Sum Rule, the derivative of $2 \sqrt[3]{8 + h} - 2 \cdot 2$ with respect to $h$ is $\frac{d}{d h} [2 \sqrt[3]{8 + h}] + \frac{d}{d h} [- 2 \cdot 2]$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [2 \sqrt[3]{8 + h}] + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5

Evaluate $\frac{d}{d h} [2 \sqrt[3]{8 + h}]$ .

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Step 1.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8 + h}$ as ${(8 + h)}^{\frac{1}{3}}$ .

$lim_{h \to 0} \frac{\frac{d}{d h} [2 {(8 + h)}^{\frac{1}{3}}] + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.2

Since $2$ is constant with respect to $h$ , the derivative of $2 {(8 + h)}^{\frac{1}{3}}$ with respect to $h$ is $2 \frac{d}{d h} [{(8 + h)}^{\frac{1}{3}}]$ .

$lim_{h \to 0} \frac{2 \frac{d}{d h} [{(8 + h)}^{\frac{1}{3}}] + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.3

Differentiate using the chain rule, which states that $\frac{d}{d h} [f (g (h))]$ is $f' (g (h)) g' (h)$ where $f (h) = h^{\frac{1}{3}}$ and $g (h) = 8 + h$ .

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Step 1.3.5.3.1

To apply the Chain Rule, set $u$ as $8 + h$ .

$lim_{h \to 0} \frac{2 (\frac{d}{d u} [u^{\frac{1}{3}}] \frac{d}{d h} [8 + h]) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.3.2

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

$lim_{h \to 0} \frac{2 (\frac{1}{3} u^{\frac{1}{3} - 1} \frac{d}{d h} [8 + h]) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.3.3

Replace all occurrences of $u$ with $8 + h$ .

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1}{3} - 1} \frac{d}{d h} [8 + h]) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.4

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

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1}{3} - 1} (\frac{d}{d h} [8] + \frac{d}{d h} [h])) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.5

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

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1}{3} - 1} (0 + \frac{d}{d h} [h])) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.6

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

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1}{3} - 1} (0 + 1)) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.7

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1}{3} - 1 \cdot \frac{3}{3}} (0 + 1)) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.8

Combine $- 1$ and $\frac{3}{3}$ .

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}} (0 + 1)) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.9

Combine the numerators over the common denominator.

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1 - 1 \cdot 3}{3}} (0 + 1)) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.10

Simplify the numerator.

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Step 1.3.5.10.1

Multiply $- 1$ by $3$ .

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{1 - 3}{3}} (0 + 1)) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.10.2

Subtract $3$ from $1$ .

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{\frac{- 2}{3}} (0 + 1)) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.11

Move the negative in front of the fraction.

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{- \frac{2}{3}} (0 + 1)) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.12

Add $0$ and $1$ .

$lim_{h \to 0} \frac{2 (\frac{1}{3} {(8 + h)}^{- \frac{2}{3}} \cdot 1) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.13

Combine $\frac{1}{3}$ and ${(8 + h)}^{- \frac{2}{3}}$ .

$lim_{h \to 0} \frac{2 (\frac{{(8 + h)}^{- \frac{2}{3}}}{3} \cdot 1) + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.14

Multiply $\frac{{(8 + h)}^{- \frac{2}{3}}}{3}$ by $1$ .

$lim_{h \to 0} \frac{2 \frac{{(8 + h)}^{- \frac{2}{3}}}{3} + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.15

Move ${(8 + h)}^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{h \to 0} \frac{2 \frac{1}{3 {(8 + h)}^{\frac{2}{3}}} + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.5.16

Combine $2$ and $\frac{1}{3 {(8 + h)}^{\frac{2}{3}}}$ .

$lim_{h \to 0} \frac{\frac{2}{3 {(8 + h)}^{\frac{2}{3}}} + \frac{d}{d h} [- 2 \cdot 2]}{\frac{d}{d h} [h]}$

Step 1.3.6

Evaluate $\frac{d}{d h} [- 2 \cdot 2]$ .

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Step 1.3.6.1

Multiply $- 2$ by $2$ .

$lim_{h \to 0} \frac{\frac{2}{3 {(8 + h)}^{\frac{2}{3}}} + \frac{d}{d h} [- 4]}{\frac{d}{d h} [h]}$

Step 1.3.6.2

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

$lim_{h \to 0} \frac{\frac{2}{3 {(8 + h)}^{\frac{2}{3}}} + 0}{\frac{d}{d h} [h]}$

Step 1.3.7

Add $\frac{2}{3 {(8 + h)}^{\frac{2}{3}}}$ and $0$ .

$lim_{h \to 0} \frac{\frac{2}{3 {(8 + h)}^{\frac{2}{3}}}}{\frac{d}{d h} [h]}$

Step 1.3.8

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

$lim_{h \to 0} \frac{\frac{2}{3 {(8 + h)}^{\frac{2}{3}}}}{1}$

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{h \to 0} \frac{2}{3 {(8 + h)}^{\frac{2}{3}}} \cdot 1$

Step 1.5

Multiply $\frac{2}{3 {(8 + h)}^{\frac{2}{3}}}$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{2}{3}$ outside of the limit because it is constant with respect to $h$ .

$\frac{2}{3} lim_{h \to 0} \frac{1}{{(8 + h)}^{\frac{2}{3}}}$

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{lim_{h \to 0} 1}{lim_{h \to 0} {(8 + h)}^{\frac{2}{3}}}$

Step 2.3

Evaluate the limit of $1$ which is constant as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{1}{lim_{h \to 0} {(8 + h)}^{\frac{2}{3}}}$

Step 2.4

Move the exponent $\frac{2}{3}$ from ${(8 + h)}^{\frac{2}{3}}$ outside the limit using the Limits Power Rule.

$\frac{2}{3} \cdot \frac{1}{{(lim_{h \to 0} 8 + h)}^{\frac{2}{3}}}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{1}{{(lim_{h \to 0} 8 + lim_{h \to 0} h)}^{\frac{2}{3}}}$

Step 2.6

Evaluate the limit of $8$ which is constant as $h$ approaches $0$ .

$\frac{2}{3} \cdot \frac{1}{{(8 + lim_{h \to 0} h)}^{\frac{2}{3}}}$

Step 3

Evaluate the limit of $h$ by plugging in $0$ for $h$ .

$\frac{2}{3} \cdot \frac{1}{{(8 + 0)}^{\frac{2}{3}}}$

Step 4

Simplify the answer.

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Step 4.1

Combine.

$\frac{2 \cdot 1}{3 {(8 + 0)}^{\frac{2}{3}}}$

Step 4.2

Multiply $2$ by $1$ .

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

Add $8$ and $0$ .

$\frac{2}{3 \cdot 8^{\frac{2}{3}}}$

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Cancel the common factor of $3$ .

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Step 4.3.4.1

Cancel the common factor.

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

Step 4.3.4.2

Rewrite the expression.

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

Step 4.3.5

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

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

Step 4.4

Multiply $3$ by $4$ .

$\frac{2}{12}$

Step 4.5

Cancel the common factor of $2$ and $12$ .

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Step 4.5.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{12}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $2$ out of $12$ .

$\frac{2 \cdot 1}{2 \cdot 6}$

Step 4.5.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 6}$

Step 4.5.2.3

Rewrite the expression.

$\frac{1}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{6}$

Decimal Form:

$0.1\overline{6}$