Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

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

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

Step 1.2.2

Move the limit under the radical sign.

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

Step 1.2.3

Move the limit inside the logarithm.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

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

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

Step 1.2.5

Simplify the answer.

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Step 1.2.5.1

Any root of $1$ is $1$ .

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

Step 1.2.5.2

Multiply $\ln (1)$ by $1$ .

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

Step 1.2.5.3

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

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

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

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

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

Step 1.3.1.2

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

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

Step 1.3.1.3

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

Simplify each term.

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Step 1.3.3.1.1

One to any power is one.

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

Step 1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

Step 1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.4

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

Undefined

$\frac{0}{0}$

Step 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.

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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

Step 3.3

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 $f (x) = x^{\frac{1}{2}}$ and $g (x) = \ln (x)$ .

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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Step 3.7.1

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

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Step 3.7.1.1

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

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

Step 3.7.1.2

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

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

Step 3.7.2

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

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

Step 3.7.3

Combine the numerators over the common denominator.

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

Step 3.7.4

Subtract $1$ from $2$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Combine the numerators over the common denominator.

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

Step 3.12

Simplify the numerator.

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Step 3.12.1

Multiply $- 1$ by $2$ .

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

Step 3.12.2

Subtract $2$ from $1$ .

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

Step 3.13

Move the negative in front of the fraction.

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

Step 3.14

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

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

Step 3.15

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

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

Step 3.16

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

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

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

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

Step 3.20

Add $2 x$ and $0$ .

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

Step 4

Convert fractional exponents to radicals.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Combine terms.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Move $2$ to the left of $\sqrt{x}$ .

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

Step 5.3

Combine the numerators over the common denominator.

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

Step 6

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 7

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

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

Step 8

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Move the limit inside the logarithm.

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

Step 13

Move the limit under the radical sign.

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

$\frac{1}{2} \cdot \frac{\frac{1}{2} \cdot \frac{2 + \ln (1)}{\sqrt{1}}}{1}$

Step 15

Simplify the answer.

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Step 15.1

Divide $\frac{1}{2} \cdot \frac{2 + \ln (1)}{\sqrt{1}}$ by $1$ .

$\frac{1}{2} (\frac{1}{2} \cdot \frac{2 + \ln (1)}{\sqrt{1}})$

Step 15.2

Simplify the numerator.

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Step 15.2.1

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

$\frac{1}{2} (\frac{1}{2} \cdot \frac{2 + 0}{\sqrt{1}})$

Step 15.2.2

Add $2$ and $0$ .

$\frac{1}{2} (\frac{1}{2} \cdot \frac{2}{\sqrt{1}})$

Step 15.3

Any root of $1$ is $1$ .

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

Step 15.4

Cancel the common factor of $2$ .

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Step 15.4.1

Cancel the common factor.

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

Step 15.4.2

Rewrite the expression.

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

Step 15.5

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

$\frac{1}{2}$