Calculus Examples

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

Step 1

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

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

Step 2

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

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

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 $x$ .

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

Step 3.2

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

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

Step 4

Simplify the answer.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (\frac{π}{4}) \frac{4}{π}$

Step 4.2

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

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

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Factor $2$ out of $4$ .

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

Step 4.3.2

Cancel the common factor.

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

Step 4.3.3

Rewrite the expression.

$\sqrt{2} \frac{2}{π}$

Step 4.4

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

$\frac{\sqrt{2} \cdot 2}{π}$

Step 4.5

Move $2$ to the left of $\sqrt{2}$ .

$\frac{2 \sqrt{2}}{π}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \sqrt{2}}{π}$

Decimal Form:

$0.90031631 \dots$