Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 10

Simplify the answer.

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Step 10.1

Multiply the numerator and denominator of the fraction by $4$ .

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Step 10.1.1

Multiply $\frac{\sin (\frac{x}{4} + \frac{π}{4}) - 1 \cdot 1}{\frac{x}{4} - \frac{π}{4}}$ by $\frac{4}{4}$ .

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

Step 10.1.2

Combine.

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

Step 10.2

Apply the distributive property.

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

Step 10.3

Simplify by cancelling.

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Step 10.3.1

Cancel the common factor of $4$ .

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Step 10.3.1.1

Cancel the common factor.

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

Step 10.3.1.2

Rewrite the expression.

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

Step 10.3.2

Cancel the common factor of $4$ .

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Step 10.3.2.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

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

Step 10.3.2.2

Cancel the common factor.

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

Step 10.3.2.3

Rewrite the expression.

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

Step 10.4

Simplify the numerator.

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Step 10.4.1

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

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Step 10.4.1.1

Factor $4$ out of $4 \sin (\frac{x}{4} + \frac{π}{4})$ .

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

Step 10.4.1.2

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

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

Step 10.4.2

Multiply $- 1$ by $1$ .

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