Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the answer.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (4 \frac{π}{2}) \frac{2}{π}$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $4$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\sin (2 π) \frac{2}{π}$

Step 5.3

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sin (0) \frac{2}{π}$

Step 5.4

The exact value of $\sin (0)$ is $0$ .

$0 \frac{2}{π}$

Step 5.5

Multiply $0$ by $\frac{2}{π}$ .

$0$