Calculus Examples

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

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.

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

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 $\frac{π}{4}$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

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 $x$ .

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

Step 1.1.2.4.2

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

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

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

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

Step 1.1.2.5.1.2

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

$\frac{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}}{lim_{x \to \frac{π}{4}} \tan (x) - 1}$

Step 1.1.2.5.2

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{2} - \sqrt{2}}{2}}{lim_{x \to \frac{π}{4}} \tan (x) - 1}$

Step 1.1.2.5.3

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

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

Step 1.1.2.5.4

Divide $0$ by $2$ .

$\frac{0}{lim_{x \to \frac{π}{4}} \tan (x) - 1}$

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

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

$\frac{0}{lim_{x \to \frac{π}{4}} \tan (x) - lim_{x \to \frac{π}{4}} 1}$

Step 1.1.3.1.2

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

$\frac{0}{\tan (lim_{x \to \frac{π}{4}} x) - lim_{x \to \frac{π}{4}} 1}$

Step 1.1.3.1.3

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

$\frac{0}{\tan (lim_{x \to \frac{π}{4}} x) - 1 \cdot 1}$

Step 1.1.3.2

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

$\frac{0}{\tan (\frac{π}{4}) - 1 \cdot 1}$

Step 1.1.3.3

Simplify the answer.

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Step 1.1.3.3.1

Simplify each term.

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Step 1.1.3.3.1.1

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

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

Step 1.1.3.3.1.2

Multiply $- 1$ by $1$ .

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

Step 1.1.3.3.2

Subtract $1$ from $1$ .

$\frac{0}{0}$

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.

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

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

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

Step 1.3.2

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

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

Step 1.3.3

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

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

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

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

Step 1.3.4.2

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

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

Step 1.3.4.3

Multiply $- 1$ by $- 1$ .

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

Step 1.3.4.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.3.7

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

$lim_{x \to \frac{π}{4}} \frac{\cos (x) + \sin (x)}{\sec^{2} (x) + 0}$

Step 1.3.8

Simplify.

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Step 1.3.8.1

Add $\sec^{2} (x)$ and $0$ .

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

Step 1.3.8.2

Rewrite $\sec (x)$ in terms of sines and cosines.

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

Step 1.3.8.3

Apply the product rule to $\frac{1}{\cos (x)}$ .

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

Step 1.3.8.4

One to any power is one.

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

Step 1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 2

Evaluate the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

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

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 $x$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify the answer.

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Step 4.1

Simplify each term.

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

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

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

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Cancel the common factor.

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

Step 4.4.2

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

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

Step 4.5

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

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

Step 4.6

Apply the product rule to $\frac{\sqrt{2}}{2}$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

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

Step 4.7.4.2

Rewrite the expression.

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

Step 4.7.5

Evaluate the exponent.

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

Step 4.8

Raise $2$ to the power of $2$ .

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

Step 4.9

Cancel the common factor of $2$ and $4$ .

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Step 4.9.1

Factor $2$ out of $2$ .

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

Step 4.9.2

Cancel the common factors.

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Step 4.9.2.1

Factor $2$ out of $4$ .

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

Step 4.9.2.2

Cancel the common factor.

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

Step 4.9.2.3

Rewrite the expression.

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

Step 4.10

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:

$0.70710678 \dots$