Calculus Examples

$lim_{x \to \frac{π}{4}} \frac{1 - \tan (x)}{\sin (x) - \cos (x)}$

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.

$\frac{lim_{x \to \frac{π}{4}} 1 - \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

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 $\frac{π}{4}$ .

$\frac{lim_{x \to \frac{π}{4}} 1 - lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.1.2

Evaluate the limit of $1$ which is constant as $x$ approaches $\frac{π}{4}$ .

$\frac{1 - lim_{x \to \frac{π}{4}} \tan (x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.1.3

Move the limit inside the trig function because tangent is continuous.

$\frac{1 - \tan (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{1 - \tan (\frac{π}{4})}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{1 - 1 \cdot 1}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.3.1.2

Multiply $- 1$ by $1$ .

$\frac{1 - 1}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

Step 1.1.2.3.2

Subtract $1$ from $1$ .

$\frac{0}{lim_{x \to \frac{π}{4}} \sin (x) - \cos (x)}$

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 $\frac{π}{4}$ .

$\frac{0}{lim_{x \to \frac{π}{4}} \sin (x) - lim_{x \to \frac{π}{4}} \cos (x)}$

Step 1.1.3.2

Move the limit inside the trig function because sine is continuous.

$\frac{0}{\sin (lim_{x \to \frac{π}{4}} x) - lim_{x \to \frac{π}{4}} \cos (x)}$

Step 1.1.3.3

Move the limit inside the trig function because cosine is continuous.

$\frac{0}{\sin (lim_{x \to \frac{π}{4}} x) - \cos (lim_{x \to \frac{π}{4}} x)}$

Step 1.1.3.4

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 1.1.3.4.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{0}{\sin (\frac{π}{4}) - \cos (lim_{x \to \frac{π}{4}} x)}$

Step 1.1.3.4.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{0}{\sin (\frac{π}{4}) - \cos (\frac{π}{4})}$

Step 1.1.3.5

Simplify the answer.

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Step 1.1.3.5.1

Simplify each term.

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Step 1.1.3.5.1.1

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$\frac{0}{\frac{\sqrt{2}}{2} - \cos (\frac{π}{4})}$

Step 1.1.3.5.1.2

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.1.3.5.2

Combine the numerators over the common denominator.

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

Step 1.1.3.5.3

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

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

Step 1.1.3.5.4

Divide $0$ by $2$ .

$\frac{0}{0}$

Step 1.1.3.5.5

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

Undefined

$\frac{0}{0}$

Step 1.1.3.6

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.

$lim_{x \to \frac{π}{4}} \frac{1 - \tan (x)}{\sin (x) - \cos (x)} = lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [1 - \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [1 - \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.2

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

$lim_{x \to \frac{π}{4}} \frac{\frac{d}{d x} [1] + \frac{d}{d x} [- \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.3

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

$lim_{x \to \frac{π}{4}} \frac{0 + \frac{d}{d x} [- \tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.4

Evaluate $\frac{d}{d x} [- \tan (x)]$ .

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Step 1.3.4.1

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

$lim_{x \to \frac{π}{4}} \frac{0 - \frac{d}{d x} [\tan (x)]}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.4.2

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to \frac{π}{4}} \frac{0 - \sec^{2} (x)}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.5

Subtract $\sec^{2} (x)$ from $0$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{\frac{d}{d x} [\sin (x) - \cos (x)]}$

Step 1.3.6

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

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [- \cos (x)]}$

Step 1.3.7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

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

Step 1.3.8

Evaluate $\frac{d}{d x} [- \cos (x)]$ .

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Step 1.3.8.1

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

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

Step 1.3.8.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{\cos (x) - - \sin (x)}$

Step 1.3.8.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.8.4

Multiply $\sin (x)$ by $1$ .

$lim_{x \to \frac{π}{4}} \frac{- \sec^{2} (x)}{\cos (x) + \sin (x)}$

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 $\frac{π}{4}$ .

$\frac{lim_{x \to \frac{π}{4}} - \sec^{2} (x)}{lim_{x \to \frac{π}{4}} \cos (x) + \sin (x)}$

Step 2.2

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

$\frac{- lim_{x \to \frac{π}{4}} \sec^{2} (x)}{lim_{x \to \frac{π}{4}} \cos (x) + \sin (x)}$

Step 2.3

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

$\frac{- {(lim_{x \to \frac{π}{4}} \sec (x))}^{2}}{lim_{x \to \frac{π}{4}} \cos (x) + \sin (x)}$

Step 2.4

Move the limit inside the trig function because secant is continuous.

$\frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \cos (x) + \sin (x)}$

Step 2.5

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{4}$ .

$\frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{lim_{x \to \frac{π}{4}} \cos (x) + lim_{x \to \frac{π}{4}} \sin (x)}$

Step 2.6

Move the limit inside the trig function because cosine is continuous.

$\frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{\cos (lim_{x \to \frac{π}{4}} x) + lim_{x \to \frac{π}{4}} \sin (x)}$

Step 2.7

Move the limit inside the trig function because sine is continuous.

$\frac{- \sec^{2} (lim_{x \to \frac{π}{4}} x)}{\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)}$

Step 3

Evaluate the limits by plugging in $\frac{π}{4}$ for all occurrences of $x$ .

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Step 3.1

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{- \sec^{2} (\frac{π}{4})}{\cos (lim_{x \to \frac{π}{4}} x) + \sin (lim_{x \to \frac{π}{4}} x)}$

Step 3.2

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{- \sec^{2} (\frac{π}{4})}{\cos (\frac{π}{4}) + \sin (lim_{x \to \frac{π}{4}} x)}$

Step 3.3

Evaluate the limit of $x$ by plugging in $\frac{π}{4}$ for $x$ .

$\frac{- \sec^{2} (\frac{π}{4})}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

$\frac{- {(\frac{2}{\sqrt{2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.2

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

$\frac{- {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3

Combine and simplify the denominator.

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Step 4.1.3.1

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

$\frac{- {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.2

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

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.3

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

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.4

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

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.5

Add $1$ and $1$ .

$\frac{- {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.6

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

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Step 4.1.3.6.1

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

$\frac{- {(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.6.2

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

$\frac{- {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.6.3

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

$\frac{- {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.6.4

Cancel the common factor of $2$ .

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Step 4.1.3.6.4.1

Cancel the common factor.

$\frac{- {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.6.4.2

Rewrite the expression.

$\frac{- {(\frac{2 \sqrt{2}}{2^{1}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.3.6.5

Evaluate the exponent.

$\frac{- {(\frac{2 \sqrt{2}}{2})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.4

Cancel the common factor of $2$ .

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Step 4.1.4.1

Cancel the common factor.

$\frac{- {(\frac{2 \sqrt{2}}{2})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.4.2

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

$\frac{- {\sqrt{2}}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.5

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

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Step 4.1.5.1

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

$\frac{- {(2^{\frac{1}{2}})}^{2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.5.2

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

$\frac{- 2^{\frac{1}{2} \cdot 2}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.5.3

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

$\frac{- 2^{\frac{2}{2}}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.5.4

Cancel the common factor of $2$ .

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Step 4.1.5.4.1

Cancel the common factor.

$\frac{- 2^{\frac{2}{2}}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.5.4.2

Rewrite the expression.

$\frac{- 2^{1}}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.1.5.5

Evaluate the exponent.

$\frac{- 1 \cdot 2}{\cos (\frac{π}{4}) + \sin (\frac{π}{4})}$

Step 4.2

Simplify the denominator.

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Step 4.2.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.2.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

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

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Step 4.2.4.1

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

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

Step 4.2.4.2

Reduce the expression by cancelling the common factors.

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Step 4.2.4.2.1

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

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Step 4.2.4.2.1.1

Cancel the common factor.

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

Step 4.2.4.2.1.2

Rewrite the expression.

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

Step 4.2.4.2.2

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

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

Step 4.3

Multiply $- 1$ by $2$ .

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

Step 4.4

Move the negative in front of the fraction.

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

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:

$- 1.41421356 \dots$