Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

Move the limit under the radical sign.

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Move the limit under the radical sign.

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

Step 1.1.2.6

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

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

Step 1.1.2.7

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

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Step 1.1.2.7.1

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

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

Step 1.1.2.7.2

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

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

Step 1.1.2.8

Simplify the answer.

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Step 1.1.2.8.1

Simplify each term.

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Step 1.1.2.8.1.1

One to any power is one.

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

Step 1.1.2.8.1.2

Any root of $1$ is $1$ .

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

Step 1.1.2.8.1.3

Any root of $1$ is $1$ .

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

Step 1.1.2.8.1.4

Multiply $- 2$ by $1$ .

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

Step 1.1.2.8.2

Subtract $2$ from $1$ .

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

Step 1.1.2.8.3

Add $- 1$ and $1$ .

$\frac{0}{lim_{x \to 1} {(x - 1)}^{2}}$

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

Move the exponent $2$ from ${(x - 1)}^{2}$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 1} x - 1)}^{2}}$

Step 1.1.3.1.2

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

$\frac{0}{{(lim_{x \to 1} x - lim_{x \to 1} 1)}^{2}}$

Step 1.1.3.1.3

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

$\frac{0}{{(lim_{x \to 1} x - 1 \cdot 1)}^{2}}$

Step 1.1.3.2

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

$\frac{0}{{(1 - 1 \cdot 1)}^{2}}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $1$ .

$\frac{0}{{(1 - 1)}^{2}}$

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0^{2}}$

Step 1.1.3.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0}$

Step 1.1.3.3.4

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

By the Sum Rule, the derivative of $\sqrt[3]{x^{2}} - 2 \sqrt[3]{x} + 1$ with respect to $x$ is $\frac{d}{d x} [\sqrt[3]{x^{2}}] + \frac{d}{d x} [- 2 \sqrt[3]{x}] + \frac{d}{d x} [1]$ .

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

Step 1.3.3

Evaluate $\frac{d}{d x} [\sqrt[3]{x^{2}}]$ .

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Step 1.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

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

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 $n = \frac{2}{3}$ .

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Combine the numerators over the common denominator.

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

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.3.3.6.2

Subtract $3$ from $2$ .

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

Step 1.3.3.7

Move the negative in front of the fraction.

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

Step 1.3.4

Evaluate $\frac{d}{d x} [- 2 \sqrt[3]{x}]$ .

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Step 1.3.4.1

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

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

Step 1.3.4.2

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x^{\frac{1}{3}}$ with respect to $x$ is $- 2 \frac{d}{d x} [x^{\frac{1}{3}}]$ .

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

Step 1.3.4.3

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

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

Step 1.3.4.4

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

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

Step 1.3.4.5

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

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

Step 1.3.4.6

Combine the numerators over the common denominator.

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

Step 1.3.4.7

Simplify the numerator.

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Step 1.3.4.7.1

Multiply $- 1$ by $3$ .

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

Step 1.3.4.7.2

Subtract $3$ from $1$ .

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

Step 1.3.4.8

Move the negative in front of the fraction.

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

Step 1.3.4.9

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

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

Step 1.3.4.10

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

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

Step 1.3.4.11

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.3.4.12

Move the negative in front of the fraction.

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

Step 1.3.5

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

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

Step 1.3.6

Simplify.

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Step 1.3.6.1

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

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

Step 1.3.6.2

Combine terms.

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Step 1.3.6.2.1

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

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

Step 1.3.6.2.2

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

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

Step 1.3.7

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

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

Step 1.3.8

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.3.8.1

Apply the distributive property.

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

Step 1.3.8.2

Apply the distributive property.

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

Step 1.3.8.3

Apply the distributive property.

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

Step 1.3.9

Simplify and combine like terms.

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Step 1.3.9.1

Simplify each term.

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Step 1.3.9.1.1

Multiply $x$ by $x$ .

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

Step 1.3.9.1.2

Move $- 1$ to the left of $x$ .

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

Step 1.3.9.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.9.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 1.3.9.1.5

Multiply $- 1$ by $- 1$ .

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

Step 1.3.9.2

Subtract $x$ from $- x$ .

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

Step 1.3.10

By the Sum Rule, the derivative of $x^{2} - 2 x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [1]$ .

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

Step 1.3.11

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

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

Step 1.3.12

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

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

Step 1.3.13

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

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

Step 1.3.14

Multiply $- 2$ by $1$ .

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

Step 1.3.15

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

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

Step 1.3.16

Add $2 x - 2$ and $0$ .

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

Step 1.4

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

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

Step 1.5

Combine terms.

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Step 1.5.1

To write $\frac{2}{3 \sqrt[3]{x}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

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

Step 1.5.2

To write $- \frac{2}{3 x^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{\sqrt[3]{x}}{\sqrt[3]{x}}$ .

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

Step 1.5.3

Write each expression with a common denominator of $3 \sqrt[3]{x} x^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.1

Multiply $\frac{2}{3 \sqrt[3]{x}}$ by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

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

Step 1.5.3.2

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

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

Step 1.5.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.5.3.4

Combine the numerators over the common denominator.

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

Step 1.5.3.5

Add $2$ and $1$ .

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

Step 1.5.3.6

Cancel the common factor of $3$ .

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Step 1.5.3.6.1

Cancel the common factor.

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

Step 1.5.3.6.2

Rewrite the expression.

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

Step 1.5.3.7

Multiply $\frac{2}{3 x^{\frac{2}{3}}}$ by $\frac{\sqrt[3]{x}}{\sqrt[3]{x}}$ .

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

Step 1.5.3.8

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

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

Step 1.5.3.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.5.3.10

Combine the numerators over the common denominator.

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

Step 1.5.3.11

Add $1$ and $2$ .

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

Step 1.5.3.12

Cancel the common factor of $3$ .

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Step 1.5.3.12.1

Cancel the common factor.

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

Step 1.5.3.12.2

Rewrite the expression.

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

Step 1.5.4

Combine the numerators over the common denominator.

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

Step 2

Evaluate the limit.

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Step 2.1

Simplify the limit argument.

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Step 2.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.1.2

Multiply $\frac{2 x^{\frac{2}{3}} - 2 \sqrt[3]{x}}{3 x^{1}}$ by $\frac{1}{2 x - 2}$ .

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

Step 2.1.3

Reduce.

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Step 2.1.3.1

Factor $2$ out of $2 x^{\frac{2}{3}}$ .

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

Step 2.1.3.2

Factor $2$ out of $- 2 \sqrt[3]{x}$ .

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

Step 2.1.3.3

Factor $2$ out of $2 (x^{\frac{2}{3}}) + 2 (- \sqrt[3]{x})$ .

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

Step 2.1.3.4

Cancel the common factors.

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Step 2.1.3.4.1

Factor $2$ out of $3 x^{1} (2 x - 2)$ .

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

Step 2.1.3.4.2

Cancel the common factor.

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

Step 2.1.3.4.3

Rewrite the expression.

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

Step 2.2

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

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

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

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

Step 3.1.2.2

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

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

Step 3.1.2.3

Move the limit under the radical sign.

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

Step 3.1.2.4

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

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Step 3.1.2.4.1

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

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

Step 3.1.2.4.2

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

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

Step 3.1.2.5

Simplify the answer.

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Step 3.1.2.5.1

Simplify each term.

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Step 3.1.2.5.1.1

One to any power is one.

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

Step 3.1.2.5.1.2

Any root of $1$ is $1$ .

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

Step 3.1.2.5.1.3

Multiply $- 1$ by $1$ .

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

Step 3.1.2.5.2

Subtract $1$ from $1$ .

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

Step 3.1.3

Evaluate the limit of the denominator.

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.3.3

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

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

Step 3.1.3.4

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

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Step 3.1.3.4.1

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

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

Step 3.1.3.4.2

Evaluate the exponent.

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

Step 3.1.3.4.3

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

$\frac{1}{3} \cdot \frac{0}{1 \cdot (1 - 1 \cdot 1)}$

Step 3.1.3.5

Simplify the answer.

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Step 3.1.3.5.1

Multiply $1 - 1 \cdot 1$ by $1$ .

$\frac{1}{3} \cdot \frac{0}{1 - 1 \cdot 1}$

Step 3.1.3.5.2

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{0}{1 - 1}$

Step 3.1.3.5.3

Subtract $1$ from $1$ .

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.1.3.5.4

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.1.3.6

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.1.4

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

Undefined

$\frac{1}{3} \cdot \frac{0}{0}$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

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

Step 3.3.2

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

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

Step 3.3.3

Evaluate $\frac{d}{d x} [x^{\frac{2}{3}}]$ .

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Step 3.3.3.1

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

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

Step 3.3.3.2

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

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

Step 3.3.3.3

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

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

Step 3.3.3.4

Combine the numerators over the common denominator.

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

Step 3.3.3.5

Simplify the numerator.

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Step 3.3.3.5.1

Multiply $- 1$ by $3$ .

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

Step 3.3.3.5.2

Subtract $3$ from $2$ .

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

Step 3.3.3.6

Move the negative in front of the fraction.

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

Step 3.3.4

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

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Step 3.3.4.1

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

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

Step 3.3.4.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{1}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{1}{3}}]$ .

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

Step 3.3.4.3

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

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

Step 3.3.4.4

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

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

Step 3.3.4.5

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

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

Step 3.3.4.6

Combine the numerators over the common denominator.

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

Step 3.3.4.7

Simplify the numerator.

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Step 3.3.4.7.1

Multiply $- 1$ by $3$ .

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

Step 3.3.4.7.2

Subtract $3$ from $1$ .

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

Step 3.3.4.8

Move the negative in front of the fraction.

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

Step 3.3.4.9

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

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

Step 3.3.4.10

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.3.5

Simplify.

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Step 3.3.5.1

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

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

Step 3.3.5.2

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

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

Step 3.3.6

Simplify.

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

Step 3.3.7

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = x - 1$ .

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

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

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

Step 3.3.11

Add $1$ and $0$ .

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

Step 3.3.12

Multiply $x$ by $1$ .

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

Step 3.3.13

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

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

Step 3.3.14

Multiply $x - 1$ by $1$ .

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

Step 3.3.15

Add $x$ and $x$ .

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

Step 3.4

Rewrite $x^{\frac{1}{3}}$ as $\sqrt[3]{x}$ .

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

Step 3.5

Combine terms.

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Step 3.5.1

To write $\frac{2}{3 \sqrt[3]{x}}$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

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

Step 3.5.2

To write $- \frac{1}{3 x^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{\sqrt[3]{x}}{\sqrt[3]{x}}$ .

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

Step 3.5.3

Write each expression with a common denominator of $3 \sqrt[3]{x} x^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.5.3.1

Multiply $\frac{2}{3 \sqrt[3]{x}}$ by $\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}}$ .

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

Step 3.5.3.2

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

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

Step 3.5.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.3.4

Combine the numerators over the common denominator.

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

Step 3.5.3.5

Add $2$ and $1$ .

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

Step 3.5.3.6

Cancel the common factor of $3$ .

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Step 3.5.3.6.1

Cancel the common factor.

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

Step 3.5.3.6.2

Rewrite the expression.

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

Step 3.5.3.7

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

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

Step 3.5.3.8

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

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

Step 3.5.3.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.3.10

Combine the numerators over the common denominator.

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

Step 3.5.3.11

Add $1$ and $2$ .

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

Step 3.5.3.12

Cancel the common factor of $3$ .

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Step 3.5.3.12.1

Cancel the common factor.

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

Step 3.5.3.12.2

Rewrite the expression.

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

Step 3.5.4

Combine the numerators over the common denominator.

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

Step 4

Evaluate the limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

Move the limit under the radical sign.

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

Step 4.8

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

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

Step 4.9

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

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

Step 4.10

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

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

Step 5

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

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Evaluate the exponent.

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

Step 5.5

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

$\frac{1}{3} \cdot \frac{\frac{1}{3} \cdot \frac{2 \cdot 1^{\frac{2}{3}} - \sqrt[3]{1}}{1}}{2 \cdot 1 - 1 \cdot 1}$

Step 6

Simplify the answer.

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Step 6.1

Divide $2 \cdot 1^{\frac{2}{3}} - \sqrt[3]{1}$ by $1$ .

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

Step 6.2

Simplify the numerator.

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Step 6.2.1

One to any power is one.

$\frac{1}{3} \cdot \frac{\frac{1}{3} (2 \cdot 1 - \sqrt[3]{1})}{2 \cdot 1 - 1 \cdot 1}$

Step 6.2.2

Multiply $2$ by $1$ .

$\frac{1}{3} \cdot \frac{\frac{1}{3} (2 - \sqrt[3]{1})}{2 \cdot 1 - 1 \cdot 1}$

Step 6.2.3

Any root of $1$ is $1$ .

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

Step 6.2.4

Multiply $- 1$ by $1$ .

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

Step 6.2.5

Subtract $1$ from $2$ .

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

Step 6.3

Simplify the denominator.

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Step 6.3.1

Multiply $2$ by $1$ .

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

Step 6.3.2

Multiply $- 1$ by $1$ .

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

Step 6.3.3

Subtract $1$ from $2$ .

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

Step 6.4

Multiply $\frac{1}{3}$ by $1$ .

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

Step 6.5

Divide $\frac{1}{3}$ by $1$ .

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

Step 6.6

Multiply $\frac{1}{3} \cdot \frac{1}{3}$ .

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Step 6.6.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

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

Step 6.6.2

Multiply $3$ by $3$ .

$\frac{1}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{9}$

Decimal Form:

$0.\overline{1}$