Calculus Examples

$lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{\sqrt{x} - 1}$

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.

$\frac{lim_{x \to 1} \sqrt[3]{x} - 1}{lim_{x \to 1} \sqrt{x} - 1}$

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 $1$ .

$\frac{lim_{x \to 1} \sqrt[3]{x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x} - 1}$

Step 1.1.2.1.2

Move the limit under the radical sign.

$\frac{\sqrt[3]{lim_{x \to 1} x} - lim_{x \to 1} 1}{lim_{x \to 1} \sqrt{x} - 1}$

Step 1.1.2.1.3

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

$\frac{\sqrt[3]{lim_{x \to 1} x} - 1 \cdot 1}{lim_{x \to 1} \sqrt{x} - 1}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

$\frac{\sqrt[3]{1} - 1 \cdot 1}{lim_{x \to 1} \sqrt{x} - 1}$

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

Any root of $1$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to 1} \sqrt{x} - 1}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to 1} \sqrt{x} - 1}$

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to 1} \sqrt{x} - 1}$

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 $1$ .

$\frac{0}{lim_{x \to 1} \sqrt{x} - lim_{x \to 1} 1}$

Step 1.1.3.1.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 1} x} - lim_{x \to 1} 1}$

Step 1.1.3.1.3

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

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

Step 1.1.3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

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

Any root of $1$ is $1$ .

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

Step 1.1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

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.

$lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{\sqrt{x} - 1} = lim_{x \to 1} \frac{\frac{d}{d x} [\sqrt[3]{x} - 1]}{\frac{d}{d x} [\sqrt{x} - 1]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [\sqrt[3]{x} - 1]}{\frac{d}{d x} [\sqrt{x} - 1]}$

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt[3]{x} - 1$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{x}] + \frac{d}{d x} [- 1]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [\sqrt[3]{x}] + \frac{d}{d x} [- 1]}{\frac{d}{d x} [\sqrt{x} - 1]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt[3]{x}]$ .

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Step 1.3.3.1

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

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

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 $n = \frac{1}{3}$ .

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Combine the numerators over the common denominator.

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

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.3.3.6.2

Subtract $3$ from $1$ .

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

Step 1.3.3.7

Move the negative in front of the fraction.

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

Step 1.3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.3.5

Simplify.

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Step 1.3.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.3.5.2

Combine terms.

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Step 1.3.5.2.1

Multiply $\frac{1}{3}$ by $\frac{1}{x^{\frac{2}{3}}}$ .

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

Step 1.3.5.2.2

Add $\frac{1}{3 x^{\frac{2}{3}}}$ and $0$ .

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

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{x} - 1$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- 1]$ .

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

Step 1.3.7

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

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.3.7.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

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

Step 1.3.7.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 1.3.7.4

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 1.3.7.5

Combine the numerators over the common denominator.

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

Step 1.3.7.6

Simplify the numerator.

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Step 1.3.7.6.1

Multiply $- 1$ by $2$ .

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

Step 1.3.7.6.2

Subtract $2$ from $1$ .

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

Step 1.3.7.7

Move the negative in front of the fraction.

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

Step 1.3.8

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 1.3.9

Simplify.

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Step 1.3.9.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.3.9.2

Combine terms.

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Step 1.3.9.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

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

Step 1.3.9.2.2

Add $\frac{1}{2 x^{\frac{1}{2}}}$ and $0$ .

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.5

Rewrite $x^{\frac{1}{2}}$ as $\sqrt{x}$ .

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

Step 1.6

Combine factors.

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Step 1.6.1

Combine $2$ and $\frac{1}{3 x^{\frac{2}{3}}}$ .

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

Step 1.6.2

Combine $\frac{2}{3 x^{\frac{2}{3}}}$ and $\sqrt{x}$ .

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

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 $x$ .

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

Step 2.2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $1$ .

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

Step 2.3

Move the limit under the radical sign.

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

Step 2.4

Move the exponent $\frac{2}{3}$ from $x^{\frac{2}{3}}$ outside the limit using the Limits Power Rule.

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

Step 3

Evaluate the limits by plugging in $1$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 3.2

Evaluate the limit of $x$ by plugging in $1$ for $x$ .

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

Step 4

Simplify the answer.

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Step 4.1

Combine.

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

Step 4.2

Any root of $1$ is $1$ .

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

Step 4.3

One to any power is one.

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

Step 4.4

Multiply $2$ by $1$ .

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

Step 4.5

Multiply $3$ by $1$ .

$\frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$