Calculus Examples

$lim_{x \to 1} \frac{4 \ln (x) + 2 \ln (x^{3})}{\ln (x) - \ln (\sqrt{x})}$

Step 1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$lim_{x \to 1} \frac{4 \ln (x) + 2 \ln (x^{3})}{\ln (\frac{x}{\sqrt{x}})}$

Step 2

Apply L'Hospital's rule.

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Step 2.1

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

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Step 2.1.1

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

$\frac{lim_{x \to 1} 4 \ln (x) + 2 \ln (x^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2

Evaluate the limit of the numerator.

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Step 2.1.2.1

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

$\frac{lim_{x \to 1} 4 \ln (x) + lim_{x \to 1} 2 \ln (x^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.2

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{4 lim_{x \to 1} \ln (x) + lim_{x \to 1} 2 \ln (x^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.3

Move the limit inside the logarithm.

$\frac{4 \ln (lim_{x \to 1} x) + lim_{x \to 1} 2 \ln (x^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.4

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

$\frac{4 \ln (lim_{x \to 1} x) + 2 lim_{x \to 1} \ln (x^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.5

Move the limit inside the logarithm.

$\frac{4 \ln (lim_{x \to 1} x) + 2 \ln (lim_{x \to 1} x^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.6

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

$\frac{4 \ln (lim_{x \to 1} x) + 2 \ln ({(lim_{x \to 1} x)}^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.7

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

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Step 2.1.2.7.1

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

$\frac{4 \ln (1) + 2 \ln ({(lim_{x \to 1} x)}^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.7.2

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

$\frac{4 \ln (1) + 2 \ln (1^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.8

Simplify the answer.

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Step 2.1.2.8.1

Simplify each term.

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Step 2.1.2.8.1.1

The natural logarithm of $1$ is $0$ .

$\frac{4 \cdot 0 + 2 \ln (1^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.8.1.2

Multiply $4$ by $0$ .

$\frac{0 + 2 \ln (1^{3})}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.8.1.3

One to any power is one.

$\frac{0 + 2 \ln (1)}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.8.1.4

The natural logarithm of $1$ is $0$ .

$\frac{0 + 2 \cdot 0}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.8.1.5

Multiply $2$ by $0$ .

$\frac{0 + 0}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.2.8.2

Add $0$ and $0$ .

$\frac{0}{lim_{x \to 1} \ln (\frac{x}{\sqrt{x}})}$

Step 2.1.3

Evaluate the limit of the denominator.

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Step 2.1.3.1

Move the limit inside the logarithm.

$\frac{0}{\ln (lim_{x \to 1} \frac{x}{\sqrt{x}})}$

Step 2.1.3.2

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

$\frac{0}{\ln (\frac{lim_{x \to 1} x}{lim_{x \to 1} \sqrt{x}})}$

Step 2.1.3.3

Move the limit under the radical sign.

$\frac{0}{\ln (\frac{lim_{x \to 1} x}{\sqrt{lim_{x \to 1} x}})}$

Step 2.1.3.4

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

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Step 2.1.3.4.1

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

$\frac{0}{\ln (\frac{1}{\sqrt{lim_{x \to 1} x}})}$

Step 2.1.3.4.2

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

$\frac{0}{\ln (\frac{1}{\sqrt{1}})}$

Step 2.1.3.5

Simplify the answer.

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Step 2.1.3.5.1

Any root of $1$ is $1$ .

$\frac{0}{\ln (\frac{1}{1})}$

Step 2.1.3.5.2

Divide $1$ by $1$ .

$\frac{0}{\ln (1)}$

Step 2.1.3.5.3

The natural logarithm of $1$ is $0$ .

$\frac{0}{0}$

Step 2.1.3.5.4

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

Undefined

$\frac{0}{0}$

Step 2.1.3.6

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

Undefined

$\frac{0}{0}$

Step 2.1.4

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

Undefined

$\frac{0}{0}$

Step 2.2

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.

$lim_{x \to 1} \frac{4 \ln (x) + 2 \ln (x^{3})}{\ln (\frac{x}{\sqrt{x}})} = lim_{x \to 1} \frac{\frac{d}{d x} [4 \ln (x) + 2 \ln (x^{3})]}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3

Find the derivative of the numerator and denominator.

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Step 2.3.1

Differentiate the numerator and denominator.

$lim_{x \to 1} \frac{\frac{d}{d x} [4 \ln (x) + 2 \ln (x^{3})]}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.2

By the Sum Rule, the derivative of $4 \ln (x) + 2 \ln (x^{3})$ with respect to $x$ is $\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [2 \ln (x^{3})]$ .

$lim_{x \to 1} \frac{\frac{d}{d x} [4 \ln (x)] + \frac{d}{d x} [2 \ln (x^{3})]}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.3

Evaluate $\frac{d}{d x} [4 \ln (x)]$ .

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Step 2.3.3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 \ln (x)$ with respect to $x$ is $4 \frac{d}{d x} [\ln (x)]$ .

$lim_{x \to 1} \frac{4 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [2 \ln (x^{3})]}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.3.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$lim_{x \to 1} \frac{4 \frac{1}{x} + \frac{d}{d x} [2 \ln (x^{3})]}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.3.3

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

$lim_{x \to 1} \frac{\frac{4}{x} + \frac{d}{d x} [2 \ln (x^{3})]}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4

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

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Step 2.3.4.1

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

$lim_{x \to 1} \frac{\frac{4}{x} + 2 \frac{d}{d x} [\ln (x^{3})]}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.2

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

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Step 2.3.4.2.1

To apply the Chain Rule, set $u$ as $x^{3}$ .

$lim_{x \to 1} \frac{\frac{4}{x} + 2 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{3}])}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.2.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$lim_{x \to 1} \frac{\frac{4}{x} + 2 (\frac{1}{u} \frac{d}{d x} [x^{3}])}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.2.3

Replace all occurrences of $u$ with $x^{3}$ .

$lim_{x \to 1} \frac{\frac{4}{x} + 2 (\frac{1}{x^{3}} \frac{d}{d x} [x^{3}])}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.3

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

$lim_{x \to 1} \frac{\frac{4}{x} + 2 (\frac{1}{x^{3}} (3 x^{2}))}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.4

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

$lim_{x \to 1} \frac{\frac{4}{x} + 2 (\frac{3}{x^{3}} x^{2})}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.5

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

$lim_{x \to 1} \frac{\frac{4}{x} + 2 \frac{3 x^{2}}{x^{3}}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.6

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 2.3.4.6.1

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

$lim_{x \to 1} \frac{\frac{4}{x} + 2 \frac{x^{2} \cdot 3}{x^{3}}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.6.2

Cancel the common factors.

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Step 2.3.4.6.2.1

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

$lim_{x \to 1} \frac{\frac{4}{x} + 2 \frac{x^{2} \cdot 3}{x^{2} x}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.6.2.2

Cancel the common factor.

$lim_{x \to 1} \frac{\frac{4}{x} + 2 \frac{x^{2} \cdot 3}{x^{2} x}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.6.2.3

Rewrite the expression.

$lim_{x \to 1} \frac{\frac{4}{x} + 2 \frac{3}{x}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.7

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

$lim_{x \to 1} \frac{\frac{4}{x} + \frac{2 \cdot 3}{x}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.4.8

Multiply $2$ by $3$ .

$lim_{x \to 1} \frac{\frac{4}{x} + \frac{6}{x}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.5

Combine terms.

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Step 2.3.5.1

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{4 + 6}{x}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.5.2

Add $4$ and $6$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (\frac{x}{\sqrt{x}})]}$

Step 2.3.6

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (\frac{x}{x^{\frac{1}{2}}})]}$

Step 2.3.7

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (x \cdot x^{- \frac{1}{2}})]}$

Step 2.3.8

Multiply $x$ by $x^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.3.8.1

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

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Step 2.3.8.1.1

Raise $x$ to the power of $1$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (x^{1} x^{- \frac{1}{2}})]}$

Step 2.3.8.1.2

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (x^{1 - \frac{1}{2}})]}$

Step 2.3.8.2

Write $1$ as a fraction with a common denominator.

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (x^{\frac{2}{2} - \frac{1}{2}})]}$

Step 2.3.8.3

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (x^{\frac{2 - 1}{2}})]}$

Step 2.3.8.4

Subtract $1$ from $2$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d x} [\ln (x^{\frac{1}{2}})]}$

Step 2.3.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = x^{\frac{1}{2}}$ .

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Step 2.3.9.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{2}}$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [x^{\frac{1}{2}}]}$

Step 2.3.9.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{u} \frac{d}{d x} [x^{\frac{1}{2}}]}$

Step 2.3.9.3

Replace all occurrences of $u$ with $x^{\frac{1}{2}}$ .

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Step 2.3.9.3.1

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

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Step 2.3.9.3.1.1

Raise $x$ to the power of $1$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{1} x^{- \frac{1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}]}$

Step 2.3.9.3.1.2

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{1 - \frac{1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}]}$

Step 2.3.9.3.2

Write $1$ as a fraction with a common denominator.

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{2}{2} - \frac{1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}]}$

Step 2.3.9.3.3

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{2 - 1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}]}$

Step 2.3.9.3.4

Subtract $1$ from $2$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} \frac{d}{d x} [x^{\frac{1}{2}}]}$

Step 2.3.10

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} - 1})}$

Step 2.3.11

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}})}$

Step 2.3.12

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}})}$

Step 2.3.13

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}})}$

Step 2.3.14

Simplify the numerator.

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Step 2.3.14.1

Multiply $- 1$ by $2$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{1 - 2}{2}})}$

Step 2.3.14.2

Subtract $2$ from $1$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} (\frac{1}{2} x^{\frac{- 1}{2}})}$

Step 2.3.15

Move the negative in front of the fraction.

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} (\frac{1}{2} x^{- \frac{1}{2}})}$

Step 2.3.16

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{x^{\frac{1}{2}}} \cdot \frac{x^{- \frac{1}{2}}}{2}}$

Step 2.3.17

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{x^{- \frac{1}{2}}}{x^{\frac{1}{2}} \cdot 2}}$

Step 2.3.18

Move $2$ to the left of $x^{\frac{1}{2}}$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{x^{- \frac{1}{2}}}{2 \cdot x^{\frac{1}{2}}}}$

Step 2.3.19

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{2 x^{\frac{1}{2}} x^{\frac{1}{2}}}}$

Step 2.3.20

Simplify the denominator.

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Step 2.3.20.1

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

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Step 2.3.20.1.1

Move $x^{\frac{1}{2}}$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{2 (x^{\frac{1}{2}} x^{\frac{1}{2}})}}$

Step 2.3.20.1.2

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

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{2 x^{\frac{1}{2} + \frac{1}{2}}}}$

Step 2.3.20.1.3

Combine the numerators over the common denominator.

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{2 x^{\frac{1 + 1}{2}}}}$

Step 2.3.20.1.4

Add $1$ and $1$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{2 x^{\frac{2}{2}}}}$

Step 2.3.20.1.5

Divide $2$ by $2$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{2 x^{1}}}$

Step 2.3.20.2

Simplify $2 x^{1}$ .

$lim_{x \to 1} \frac{\frac{10}{x}}{\frac{1}{2 x}}$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 1} \frac{10}{x} (2 x)$

Step 2.5

Combine factors.

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Step 2.5.1

Combine $2$ and $\frac{10}{x}$ .

$lim_{x \to 1} \frac{2 \cdot 10}{x} x$

Step 2.5.2

Multiply $2$ by $10$ .

$lim_{x \to 1} \frac{20}{x} x$

Step 2.5.3

Combine $\frac{20}{x}$ and $x$ .

$lim_{x \to 1} \frac{20 x}{x}$

Step 2.6

Cancel the common factor of $x$ .

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Step 2.6.1

Cancel the common factor.

$lim_{x \to 1} \frac{20 x}{x}$

Step 2.6.2

Divide $20$ by $1$ .

$lim_{x \to 1} 20$

Step 3

Evaluate the limit of $20$ which is constant as $x$ approaches $1$ .

$20$