Calculus Examples

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

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.

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

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

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

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Cancel the common factor of $2$ .

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Step 1.1.2.3.1.1

Factor $2$ out of $4$ .

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

Step 1.1.2.3.1.2

Cancel the common factor.

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

Step 1.1.2.3.1.3

Rewrite the expression.

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

Step 1.1.2.3.2

The exact value of $\cos (\frac{π}{2})$ is $0$ .

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

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 $\frac{π}{4}$ .

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

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 $x$ .

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

Step 1.1.3.4.2

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

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

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 $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

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

Step 1.1.3.5.1.2

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

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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Step 1.3.2.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 1.3.2.2

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

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

Step 1.3.2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $2$ by $- 1$ .

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

Step 1.3.5

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

$lim_{x \to \frac{π}{4}} \frac{- 2 \sin (2 x) \cdot 1}{\frac{d}{d x} [\cos (x) - \sin (x)]}$

Step 1.3.6

Multiply $- 2$ by $1$ .

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

Step 1.3.7

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

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

Step 1.3.8

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

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

Step 1.3.9

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

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Step 1.3.9.1

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

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

Step 1.3.9.2

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

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

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 $x$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Cancel the common factor of $2$ .

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Step 4.1.1.1

Factor $2$ out of $4$ .

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

Step 4.1.1.2

Cancel the common factor.

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

Step 4.1.1.3

Rewrite the expression.

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

Step 4.1.2

The exact value of $\sin (\frac{π}{2})$ is $1$ .

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

Factor $- 1$ out of $- \sin (\frac{π}{4}) - \cos (\frac{π}{4})$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Combine the numerators over the common denominator.

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

Step 4.2.5

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

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Step 4.2.5.1

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

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

Step 4.2.5.2

Reduce the expression by cancelling the common factors.

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Step 4.2.5.2.1

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

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Step 4.2.5.2.1.1

Cancel the common factor.

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

Step 4.2.5.2.1.2

Rewrite the expression.

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

Step 4.2.5.2.2

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

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

Step 4.3

Cancel the common factor of $1$ and $- 1$ .

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Step 4.3.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.3.2

Move the negative in front of the fraction.

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

Step 4.4

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

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

Step 4.5

Combine and simplify the denominator.

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.5.3

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

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

Step 4.5.4

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

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

Step 4.5.5

Add $1$ and $1$ .

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

Step 4.5.6

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

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Step 4.5.6.1

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

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

Step 4.5.6.2

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

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

Step 4.5.6.3

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

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

Step 4.5.6.4

Cancel the common factor of $2$ .

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Step 4.5.6.4.1

Cancel the common factor.

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

Step 4.5.6.4.2

Rewrite the expression.

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

Step 4.5.6.5

Evaluate the exponent.

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

Step 4.6

Cancel the common factor of $2$ .

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Step 4.6.1

Move the leading negative in $- \frac{\sqrt{2}}{2}$ into the numerator.

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

Step 4.6.2

Factor $2$ out of $- 2$ .

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

Step 4.6.3

Cancel the common factor.

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

Step 4.6.4

Rewrite the expression.

$- - \sqrt{2}$

Step 4.7

Multiply $- 1$ by $- 1$ .

$1 \sqrt{2}$

Step 4.8

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

$\sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\sqrt{2}$

Decimal Form:

$1.41421356 \dots$