Calculus Examples

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

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.

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

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 $1$ .

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

Step 1.1.2.2

Move the limit under the radical sign.

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

Step 1.1.2.3

Move the limit under the radical sign.

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

Step 1.1.2.4

Evaluate the limit of $2$ which is constant as $x$ approaches $1$ .

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

Step 1.1.2.5

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

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Step 1.1.2.5.1

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

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

Step 1.1.2.5.2

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

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

Step 1.1.2.6

Simplify the answer.

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Step 1.1.2.6.1

Simplify each term.

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Step 1.1.2.6.1.1

Any root of $1$ is $1$ .

$\frac{1 + \sqrt{1} - 1 \cdot 2}{lim_{x \to 1} x - 1}$

Step 1.1.2.6.1.2

Any root of $1$ is $1$ .

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

Step 1.1.2.6.1.3

Multiply $- 1$ by $2$ .

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

Step 1.1.2.6.2

Add $1$ and $1$ .

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

Step 1.1.2.6.3

Subtract $2$ from $2$ .

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

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 $1$ .

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

Step 1.1.3.1.2

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Multiply $- 1$ by $1$ .

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

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

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 $x^{\frac{1}{3}}$ .

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

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

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

Step 1.3.3.3

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

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

Step 1.3.3.4

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

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

Step 1.3.3.5

Combine the numerators over the common denominator.

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

Step 1.3.3.6

Simplify the numerator.

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Step 1.3.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.3.3.6.2

Subtract $3$ from $1$ .

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

Step 1.3.3.7

Move the negative in front of the fraction.

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

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

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

Step 1.3.4.4

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

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

Step 1.3.4.5

Combine the numerators over the common denominator.

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

Step 1.3.4.6

Simplify the numerator.

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Step 1.3.4.6.1

Multiply $- 1$ by $2$ .

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

Step 1.3.4.6.2

Subtract $2$ from $1$ .

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

Step 1.3.4.7

Move the negative in front of the fraction.

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

Step 1.3.5

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

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

Step 1.3.6

Simplify.

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Step 1.3.6.1

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

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

Step 1.3.6.2

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

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

Step 1.3.6.3

Combine terms.

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Step 1.3.6.3.1

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

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

Step 1.3.6.3.2

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

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

Step 1.3.6.3.3

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

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

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

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

Step 1.3.10

Add $1$ and $0$ .

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

Step 1.4

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

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

Step 1.5

Combine terms.

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Write each expression with a common denominator of $6 \sqrt{x} x^{\frac{2}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.1

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

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

Step 1.5.3.2

Multiply $2$ by $3$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{2}{3}} \sqrt{x}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.3

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

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 (x^{\frac{1}{2}} x^{\frac{2}{3}})} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.4

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

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{1}{2} + \frac{2}{3}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.5

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.6

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{1}{2} \cdot \frac{3}{3} + \frac{2}{3} \cdot \frac{2}{2}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.7.1

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

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{3}{2 \cdot 3} + \frac{2}{3} \cdot \frac{2}{2}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.7.2

Multiply $2$ by $3$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{3}{6} + \frac{2}{3} \cdot \frac{2}{2}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.7.3

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{3}{6} + \frac{2 \cdot 2}{3 \cdot 2}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.7.4

Multiply $3$ by $2$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{3}{6} + \frac{2 \cdot 2}{6}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.8

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{3 + 2 \cdot 2}{6}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.9

Simplify the numerator.

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Step 1.5.3.9.1

Multiply $2$ by $2$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{3 + 4}{6}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.9.2

Add $3$ and $4$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{1}{2 \sqrt{x}} \cdot \frac{3 x^{\frac{2}{3}}}{3 x^{\frac{2}{3}}}}{1}$

Step 1.5.3.10

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

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{2 \sqrt{x} (3 x^{\frac{2}{3}})}}{1}$

Step 1.5.3.11

Multiply $3$ by $2$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 \sqrt{x} x^{\frac{2}{3}}}}{1}$

Step 1.5.3.12

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

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 (x^{\frac{2}{3}} x^{\frac{1}{2}})}}{1}$

Step 1.5.3.13

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

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2}{3} + \frac{1}{2}}}}{1}$

Step 1.5.3.14

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2}}}}{1}$

Step 1.5.3.15

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2}{3} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{3}{3}}}}{1}$

Step 1.5.3.16

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.3.16.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2 \cdot 2}{3 \cdot 2} + \frac{1}{2} \cdot \frac{3}{3}}}}{1}$

Step 1.5.3.16.2

Multiply $3$ by $2$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3}}}}{1}$

Step 1.5.3.16.3

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

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2 \cdot 2}{6} + \frac{3}{2 \cdot 3}}}}{1}$

Step 1.5.3.16.4

Multiply $2$ by $3$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2 \cdot 2}{6} + \frac{3}{6}}}}{1}$

Step 1.5.3.17

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{2 \cdot 2 + 3}{6}}}}{1}$

Step 1.5.3.18

Simplify the numerator.

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Step 1.5.3.18.1

Multiply $2$ by $2$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{4 + 3}{6}}}}{1}$

Step 1.5.3.18.2

Add $4$ and $3$ .

$lim_{x \to 1} \frac{\frac{2 \sqrt{x}}{6 x^{\frac{7}{6}}} + \frac{3 x^{\frac{2}{3}}}{6 x^{\frac{7}{6}}}}{1}$

Step 1.5.4

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{2 \sqrt{x} + 3 x^{\frac{2}{3}}}{6 x^{\frac{7}{6}}}}{1}$

Step 1.6

Divide $\frac{2 \sqrt{x} + 3 x^{\frac{2}{3}}}{6 x^{\frac{7}{6}}}$ by $1$ .

$lim_{x \to 1} \frac{2 \sqrt{x} + 3 x^{\frac{2}{3}}}{6 x^{\frac{7}{6}}}$

Step 2

Evaluate the limit.

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Step 2.1

Move the term $\frac{1}{6}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{6} lim_{x \to 1} \frac{2 \sqrt{x} + 3 x^{\frac{2}{3}}}{x^{\frac{7}{6}}}$

Step 2.2

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

$\frac{1}{6} \cdot \frac{lim_{x \to 1} 2 \sqrt{x} + 3 x^{\frac{2}{3}}}{lim_{x \to 1} x^{\frac{7}{6}}}$

Step 2.3

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

$\frac{1}{6} \cdot \frac{lim_{x \to 1} 2 \sqrt{x} + lim_{x \to 1} 3 x^{\frac{2}{3}}}{lim_{x \to 1} x^{\frac{7}{6}}}$

Step 2.4

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

$\frac{1}{6} \cdot \frac{2 lim_{x \to 1} \sqrt{x} + lim_{x \to 1} 3 x^{\frac{2}{3}}}{lim_{x \to 1} x^{\frac{7}{6}}}$

Step 2.5

Move the limit under the radical sign.

$\frac{1}{6} \cdot \frac{2 \sqrt{lim_{x \to 1} x} + lim_{x \to 1} 3 x^{\frac{2}{3}}}{lim_{x \to 1} x^{\frac{7}{6}}}$

Step 2.6

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{6} \cdot \frac{2 \sqrt{lim_{x \to 1} x} + 3 lim_{x \to 1} x^{\frac{2}{3}}}{lim_{x \to 1} x^{\frac{7}{6}}}$

Step 2.7

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

$\frac{1}{6} \cdot \frac{2 \sqrt{lim_{x \to 1} x} + 3 {(lim_{x \to 1} x)}^{\frac{2}{3}}}{lim_{x \to 1} x^{\frac{7}{6}}}$

Step 2.8

Move the exponent $\frac{7}{6}$ from $x^{\frac{7}{6}}$ outside the limit using the Limits Power Rule.

$\frac{1}{6} \cdot \frac{2 \sqrt{lim_{x \to 1} x} + 3 {(lim_{x \to 1} x)}^{\frac{2}{3}}}{{(lim_{x \to 1} x)}^{\frac{7}{6}}}$

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 $x$ .

$\frac{1}{6} \cdot \frac{2 \sqrt{1} + 3 {(lim_{x \to 1} x)}^{\frac{2}{3}}}{{(lim_{x \to 1} x)}^{\frac{7}{6}}}$

Step 3.2

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

$\frac{1}{6} \cdot \frac{2 \sqrt{1} + 3 \cdot 1^{\frac{2}{3}}}{{(lim_{x \to 1} x)}^{\frac{7}{6}}}$

Step 3.3

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

$\frac{1}{6} \cdot \frac{2 \sqrt{1} + 3 \cdot 1^{\frac{2}{3}}}{1^{\frac{7}{6}}}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Any root of $1$ is $1$ .

$\frac{1}{6} \cdot \frac{2 \cdot 1 + 3 \cdot 1^{\frac{2}{3}}}{1^{\frac{7}{6}}}$

Step 4.1.2

Multiply $2$ by $1$ .

$\frac{1}{6} \cdot \frac{2 + 3 \cdot 1^{\frac{2}{3}}}{1^{\frac{7}{6}}}$

Step 4.1.3

One to any power is one.

$\frac{1}{6} \cdot \frac{2 + 3 \cdot 1}{1^{\frac{7}{6}}}$

Step 4.1.4

Multiply $3$ by $1$ .

$\frac{1}{6} \cdot \frac{2 + 3}{1^{\frac{7}{6}}}$

Step 4.1.5

Add $2$ and $3$ .

$\frac{1}{6} \cdot \frac{5}{1^{\frac{7}{6}}}$

Step 4.2

One to any power is one.

$\frac{1}{6} \cdot \frac{5}{1}$

Step 4.3

Divide $5$ by $1$ .

$\frac{1}{6} \cdot 5$

Step 4.4

Combine $\frac{1}{6}$ and $5$ .

$\frac{5}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{5}{6}$

Decimal Form:

$0.8\overline{3}$