Calculus Examples

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

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.

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

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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Step 1.1.2.4.1

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

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

Step 1.1.2.4.2

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

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

Step 1.1.2.5

Simplify the answer.

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Step 1.1.2.5.1

Simplify each term.

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Step 1.1.2.5.1.1

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

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

Step 1.1.2.5.1.2

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

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

Step 1.1.2.5.2

Combine the numerators over the common denominator.

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

Step 1.1.2.5.3

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

$\frac{\frac{0}{2}}{lim_{x \to \frac{π}{4}} \cos (2 x)}$

Step 1.1.2.5.4

Divide $0$ by $2$ .

$\frac{0}{lim_{x \to \frac{π}{4}} \cos (2 x)}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

Evaluate the limit.

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Step 1.1.3.1.1

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

$\frac{0}{\cos (lim_{x \to \frac{π}{4}} 2 x)}$

Step 1.1.3.1.2

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

$\frac{0}{\cos (2 lim_{x \to \frac{π}{4}} x)}$

Step 1.1.3.2

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

$\frac{0}{\cos (2 \frac{π}{4})}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Cancel the common factor of $2$ .

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Step 1.1.3.3.1.1

Factor $2$ out of $4$ .

$\frac{0}{\cos (2 \frac{π}{2 (2)})}$

Step 1.1.3.3.1.2

Cancel the common factor.

$\frac{0}{\cos (2 \frac{π}{2 \cdot 2})}$

Step 1.1.3.3.1.3

Rewrite the expression.

$\frac{0}{\cos (\frac{π}{2})}$

Step 1.1.3.3.2

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

$\frac{0}{0}$

Step 1.1.3.3.3

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

Undefined

$\frac{0}{0}$

Step 1.1.3.4

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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Step 1.3.4.1

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.4

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

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

Step 1.3.5

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 1.3.5.1

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

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

Step 1.3.5.2

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

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

Step 1.3.5.3

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

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

Step 1.3.6

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

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

Step 1.3.7

Multiply $2$ by $- 1$ .

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

Step 1.3.8

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

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

Step 1.3.9

Multiply $- 2$ by $1$ .

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

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

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

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 $x$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify the answer.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

Simplify the numerator.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

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 $\sqrt{2}$ .

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

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.

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

Step 4.2.4.2.1.2

Rewrite the expression.

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

Step 4.2.4.2.2

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

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Factor $2$ out of $4$ .

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

Step 4.3.1.2

Cancel the common factor.

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

Step 4.3.1.3

Rewrite the expression.

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

Step 4.3.2

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.70710678 \dots$