Calculus Examples

$lim_{x \to 2} \frac{x - 2}{\sqrt{x} - \sqrt{4 - x}}$

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.

$\frac{lim_{x \to 2} x - 2}{lim_{x \to 2} \sqrt{x} - \sqrt{4 - x}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} x - lim_{x \to 2} 2}{lim_{x \to 2} \sqrt{x} - \sqrt{4 - x}}$

Step 1.1.2.1.2

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

$\frac{lim_{x \to 2} x - 1 \cdot 2}{lim_{x \to 2} \sqrt{x} - \sqrt{4 - x}}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{2 - 1 \cdot 2}{lim_{x \to 2} \sqrt{x} - \sqrt{4 - x}}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Multiply $- 1$ by $2$ .

$\frac{2 - 2}{lim_{x \to 2} \sqrt{x} - \sqrt{4 - x}}$

Step 1.1.2.3.2

Subtract $2$ from $2$ .

$\frac{0}{lim_{x \to 2} \sqrt{x} - \sqrt{4 - x}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{0}{lim_{x \to 2} \sqrt{x} - lim_{x \to 2} \sqrt{4 - x}}$

Step 1.1.3.2

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 2} x} - lim_{x \to 2} \sqrt{4 - x}}$

Step 1.1.3.3

Move the limit under the radical sign.

$\frac{0}{\sqrt{lim_{x \to 2} x} - \sqrt{lim_{x \to 2} 4 - x}}$

Step 1.1.3.4

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{0}{\sqrt{lim_{x \to 2} x} - \sqrt{lim_{x \to 2} 4 - lim_{x \to 2} x}}$

Step 1.1.3.5

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{0}{\sqrt{lim_{x \to 2} x} - \sqrt{4 - lim_{x \to 2} x}}$

Step 1.1.3.6

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 1.1.3.6.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{0}{\sqrt{2} - \sqrt{4 - lim_{x \to 2} x}}$

Step 1.1.3.6.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{0}{\sqrt{2} - \sqrt{4 - 2}}$

Step 1.1.3.7

Simplify the answer.

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Step 1.1.3.7.1

Subtract $2$ from $4$ .

$\frac{0}{\sqrt{2} - \sqrt{2}}$

Step 1.1.3.7.2

Subtract $\sqrt{2}$ from $\sqrt{2}$ .

$\frac{0}{0}$

Step 1.1.3.7.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.8

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.

$lim_{x \to 2} \frac{x - 2}{\sqrt{x} - \sqrt{4 - x}} = lim_{x \to 2} \frac{\frac{d}{d x} [x - 2]}{\frac{d}{d x} [\sqrt{x} - \sqrt{4 - x}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 2} \frac{\frac{d}{d x} [x - 2]}{\frac{d}{d x} [\sqrt{x} - \sqrt{4 - x}]}$

Step 1.3.2

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

$lim_{x \to 2} \frac{\frac{d}{d x} [x] + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{x} - \sqrt{4 - x}]}$

Step 1.3.3

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

$lim_{x \to 2} \frac{1 + \frac{d}{d x} [- 2]}{\frac{d}{d x} [\sqrt{x} - \sqrt{4 - x}]}$

Step 1.3.4

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

$lim_{x \to 2} \frac{1 + 0}{\frac{d}{d x} [\sqrt{x} - \sqrt{4 - x}]}$

Step 1.3.5

Add $1$ and $0$ .

$lim_{x \to 2} \frac{1}{\frac{d}{d x} [\sqrt{x} - \sqrt{4 - x}]}$

Step 1.3.6

By the Sum Rule, the derivative of $\sqrt{x} - \sqrt{4 - x}$ with respect to $x$ is $\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- \sqrt{4 - x}]$ .

$lim_{x \to 2} \frac{1}{\frac{d}{d x} [\sqrt{x}] + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7

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

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Step 1.3.7.1

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

$lim_{x \to 2} \frac{1}{\frac{d}{d x} [x^{\frac{1}{2}}] + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7.2

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

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{\frac{1}{2} - 1} + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7.3

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

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7.4

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

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7.5

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7.6

Simplify the numerator.

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Step 1.3.7.6.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{\frac{1 - 2}{2}} + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7.6.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{\frac{- 1}{2}} + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.7.7

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} + \frac{d}{d x} [- \sqrt{4 - x}]}$

Step 1.3.8

Evaluate $\frac{d}{d x} [- \sqrt{4 - x}]$ .

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Step 1.3.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x}$ as ${(4 - x)}^{\frac{1}{2}}$ .

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

Step 1.3.8.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- {(4 - x)}^{\frac{1}{2}}$ with respect to $x$ is $- \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - \frac{d}{d x} [{(4 - x)}^{\frac{1}{2}}]}$

Step 1.3.8.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = 4 - x$ .

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Step 1.3.8.3.1

To apply the Chain Rule, set $u$ as $4 - x$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [4 - x])}$

Step 1.3.8.3.2

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

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x])}$

Step 1.3.8.3.3

Replace all occurrences of $u$ with $4 - x$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1} \frac{d}{d x} [4 - x])}$

Step 1.3.8.4

By the Sum Rule, the derivative of $4 - x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- x]$ .

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

Step 1.3.8.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1} (0 + \frac{d}{d x} [- x]))}$

Step 1.3.8.6

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

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1} (0 - \frac{d}{d x} [x]))}$

Step 1.3.8.7

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

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1} (0 - 1 \cdot 1))}$

Step 1.3.8.8

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

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} (0 - 1 \cdot 1))}$

Step 1.3.8.9

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

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

Step 1.3.8.10

Combine the numerators over the common denominator.

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{1 - 1 \cdot 2}{2}} (0 - 1 \cdot 1))}$

Step 1.3.8.11

Simplify the numerator.

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Step 1.3.8.11.1

Multiply $- 1$ by $2$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{1 - 2}{2}} (0 - 1 \cdot 1))}$

Step 1.3.8.11.2

Subtract $2$ from $1$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{\frac{- 1}{2}} (0 - 1 \cdot 1))}$

Step 1.3.8.12

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{- \frac{1}{2}} (0 - 1 \cdot 1))}$

Step 1.3.8.13

Multiply $- 1$ by $1$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{- \frac{1}{2}} (0 - 1))}$

Step 1.3.8.14

Subtract $1$ from $0$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{1}{2} {(4 - x)}^{- \frac{1}{2}} \cdot - 1)}$

Step 1.3.8.15

Combine $\frac{1}{2}$ and ${(4 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - (\frac{{(4 - x)}^{- \frac{1}{2}}}{2} \cdot - 1)}$

Step 1.3.8.16

Combine $\frac{{(4 - x)}^{- \frac{1}{2}}}{2}$ and $- 1$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - \frac{{(4 - x)}^{- \frac{1}{2}} \cdot - 1}{2}}$

Step 1.3.8.17

Move $- 1$ to the left of ${(4 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - \frac{- 1 \cdot {(4 - x)}^{- \frac{1}{2}}}{2}}$

Step 1.3.8.18

Rewrite $- 1 {(4 - x)}^{- \frac{1}{2}}$ as $- {(4 - x)}^{- \frac{1}{2}}$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - \frac{- {(4 - x)}^{- \frac{1}{2}}}{2}}$

Step 1.3.8.19

Move ${(4 - x)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - \frac{- 1}{2 {(4 - x)}^{\frac{1}{2}}}}$

Step 1.3.8.20

Move the negative in front of the fraction.

$lim_{x \to 2} \frac{1}{\frac{1}{2} x^{- \frac{1}{2}} - - \frac{1}{2 {(4 - x)}^{\frac{1}{2}}}}$

Step 1.3.8.21

Multiply $- 1$ by $- 1$ .

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

Step 1.3.8.22

Multiply $\frac{1}{2 {(4 - x)}^{\frac{1}{2}}}$ by $1$ .

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

Step 1.3.9

Simplify.

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Step 1.3.9.1

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

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

Step 1.3.9.2

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

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

Step 1.4

Convert fractional exponents to radicals.

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Step 1.4.1

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

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

Step 1.4.2

Rewrite ${(4 - x)}^{\frac{1}{2}}$ as $\sqrt{4 - x}$ .

$lim_{x \to 2} \frac{1}{\frac{1}{2 \sqrt{x}} + \frac{1}{2 \sqrt{4 - x}}}$

Step 2

Evaluate the limit.

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Step 2.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{lim_{x \to 2} 1}{lim_{x \to 2} \frac{1}{2 \sqrt{x}} + \frac{1}{2 \sqrt{4 - x}}}$

Step 2.2

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

$\frac{1}{lim_{x \to 2} \frac{1}{2 \sqrt{x}} + \frac{1}{2 \sqrt{4 - x}}}$

Step 2.3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{lim_{x \to 2} \frac{1}{2 \sqrt{x}} + lim_{x \to 2} \frac{1}{2 \sqrt{4 - x}}}$

Step 2.4

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

$\frac{1}{\frac{1}{2} lim_{x \to 2} \frac{1}{\sqrt{x}} + lim_{x \to 2} \frac{1}{2 \sqrt{4 - x}}}$

Step 2.5

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{1}{\frac{1}{2} \cdot \frac{lim_{x \to 2} 1}{lim_{x \to 2} \sqrt{x}} + lim_{x \to 2} \frac{1}{2 \sqrt{4 - x}}}$

Step 2.6

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

$\frac{1}{\frac{1}{2} \cdot \frac{1}{lim_{x \to 2} \sqrt{x}} + lim_{x \to 2} \frac{1}{2 \sqrt{4 - x}}}$

Step 2.7

Move the limit under the radical sign.

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} x}} + lim_{x \to 2} \frac{1}{2 \sqrt{4 - x}}}$

Step 2.8

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

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} x}} + \frac{1}{2} lim_{x \to 2} \frac{1}{\sqrt{4 - x}}}$

Step 2.9

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $2$ .

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} x}} + \frac{1}{2} \cdot \frac{lim_{x \to 2} 1}{lim_{x \to 2} \sqrt{4 - x}}}$

Step 2.10

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

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} x}} + \frac{1}{2} \cdot \frac{1}{lim_{x \to 2} \sqrt{4 - x}}}$

Step 2.11

Move the limit under the radical sign.

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} x}} + \frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} 4 - x}}}$

Step 2.12

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $2$ .

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} x}} + \frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} 4 - lim_{x \to 2} x}}}$

Step 2.13

Evaluate the limit of $4$ which is constant as $x$ approaches $2$ .

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{lim_{x \to 2} x}} + \frac{1}{2} \cdot \frac{1}{\sqrt{4 - lim_{x \to 2} x}}}$

Step 3

Evaluate the limits by plugging in $2$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{2}} + \frac{1}{2} \cdot \frac{1}{\sqrt{4 - lim_{x \to 2} x}}}$

Step 3.2

Evaluate the limit of $x$ by plugging in $2$ for $x$ .

$\frac{1}{\frac{1}{2} \cdot \frac{1}{\sqrt{2}} + \frac{1}{2} \cdot \frac{1}{\sqrt{4 - 2}}}$

Step 4

Simplify the answer.

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Step 4.1

Subtract $2$ from $4$ .

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

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

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

Step 4.2.2

Combine and simplify the denominator.

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Step 4.2.2.1

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

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

Step 4.2.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.2.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.2.2.4

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

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

Step 4.2.2.5

Add $1$ and $1$ .

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

Step 4.2.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.2.2.6.1

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

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

Step 4.2.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2.6.3

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

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

Step 4.2.2.6.4

Cancel the common factor of $2$ .

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Step 4.2.2.6.4.1

Cancel the common factor.

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

Step 4.2.2.6.4.2

Rewrite the expression.

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

Step 4.2.2.6.5

Evaluate the exponent.

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

Step 4.2.3

Multiply $\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 4.2.3.1

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

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

Step 4.2.3.2

Multiply $2$ by $2$ .

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{1}{\sqrt{2}}}$

Step 4.2.4

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

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}$

Step 4.2.5

Combine and simplify the denominator.

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Step 4.2.5.1

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

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}}$

Step 4.2.5.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}$

Step 4.2.5.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}$

Step 4.2.5.4

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

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}}$

Step 4.2.5.5

Add $1$ and $1$ .

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{{\sqrt{2}}^{2}}}$

Step 4.2.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.2.5.6.1

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

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}$

Step 4.2.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}$

Step 4.2.5.6.3

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

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}}}$

Step 4.2.5.6.4

Cancel the common factor of $2$ .

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Step 4.2.5.6.4.1

Cancel the common factor.

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{\frac{2}{2}}}}$

Step 4.2.5.6.4.2

Rewrite the expression.

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2^{1}}}$

Step 4.2.5.6.5

Evaluate the exponent.

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{1}{2} \cdot \frac{\sqrt{2}}{2}}$

Step 4.2.6

Multiply $\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 4.2.6.1

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

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{\sqrt{2}}{2 \cdot 2}}$

Step 4.2.6.2

Multiply $2$ by $2$ .

$\frac{1}{\frac{\sqrt{2}}{4} + \frac{\sqrt{2}}{4}}$

Step 4.2.7

Combine the numerators over the common denominator.

$\frac{1}{\frac{\sqrt{2} + \sqrt{2}}{4}}$

Step 4.2.8

Rewrite $\frac{\sqrt{2} + \sqrt{2}}{4}$ in a factored form.

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Step 4.2.8.1

Add $\sqrt{2}$ and $\sqrt{2}$ .

$\frac{1}{\frac{2 \sqrt{2}}{4}}$

Step 4.2.8.2

Reduce the expression $\frac{2 \sqrt{2}}{4}$ by cancelling the common factors.

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Step 4.2.8.2.1

Factor $2$ out of $2 \sqrt{2}$ .

$\frac{1}{\frac{2 (\sqrt{2})}{4}}$

Step 4.2.8.2.2

Factor $2$ out of $4$ .

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

Step 4.2.8.2.3

Cancel the common factor.

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

Step 4.2.8.2.4

Rewrite the expression.

$\frac{1}{\frac{\sqrt{2}}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

$1 \frac{2}{\sqrt{2}}$

Step 4.4

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

$\frac{2}{\sqrt{2}}$

Step 4.5

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

$\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

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

$\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 4.6.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.6.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 4.6.4

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

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 4.6.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.6.6.1

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

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

Step 4.6.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.6.6.3

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

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

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Step 4.6.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

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

Step 4.6.6.5

Evaluate the exponent.

$\frac{2 \sqrt{2}}{2}$

Step 4.7

Cancel the common factor of $2$ .

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Step 4.7.1

Cancel the common factor.

$\frac{2 \sqrt{2}}{2}$

Step 4.7.2

Divide $\sqrt{2}$ by $1$ .

$\sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$