Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{π}{2}$ .

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

Step 2

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

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

Step 3

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $\frac{π}{2}$ .

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

Step 4

Evaluate the limit of $\frac{π}{4}$ which is constant as $x$ approaches $\frac{π}{2}$ .

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

Step 5

Evaluate the limits by plugging in $\frac{π}{2}$ for all occurrences of $x$ .

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Step 5.1

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

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

Step 5.2

Evaluate the limit of $x$ by plugging in $\frac{π}{2}$ for $x$ .

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

Step 6

Simplify the answer.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.1

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

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

Step 6.3.2

Multiply $2$ by $2$ .

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

Step 6.4

Combine the numerators over the common denominator.

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

Step 6.5

Simplify the numerator.

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Step 6.5.1

Move $2$ to the left of $π$ .

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

Step 6.5.2

Subtract $π$ from $2 π$ .

$\sin (\frac{π}{4}) \frac{2}{π}$

Step 6.6

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

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

Step 6.7

Cancel the common factor of $2$ .

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Step 6.7.1

Cancel the common factor.

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

Step 6.7.2

Rewrite the expression.

$\sqrt{2} \frac{1}{π}$

Step 6.8

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

$\frac{\sqrt{2}}{π}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{π}$

Decimal Form:

$0.45015815 \dots$