Calculus Examples

$lim_{x \to \frac{3 x}{2}} \frac{\sin (x)}{3 x}$

Step 1

Move the term $\frac{1}{3}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{3} lim_{x \to \frac{3 x}{2}} \frac{\sin (x)}{x}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $\frac{3 x}{2}$ .

$\frac{1}{3} \cdot \frac{lim_{x \to \frac{3 x}{2}} \sin (x)}{lim_{x \to \frac{3 x}{2}} x}$

Step 3

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

$\frac{1}{3} \cdot \frac{\sin (lim_{x \to \frac{3 x}{2}} x)}{lim_{x \to \frac{3 x}{2}} x}$

Step 4

Evaluate the limits by plugging in $\frac{3 x}{2}$ for all occurrences of $x$ .

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Step 4.1

Evaluate the limit of $x$ by plugging in $\frac{3 x}{2}$ for $x$ .

$\frac{1}{3} \cdot \frac{\sin (\frac{3 x}{2})}{lim_{x \to \frac{3 x}{2}} x}$

Step 4.2

Evaluate the limit of $x$ by plugging in $\frac{3 x}{2}$ for $x$ .

$\frac{1}{3} \cdot \frac{\sin (\frac{3 x}{2})}{\frac{3 x}{2}}$

Step 5

Simplify the answer.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{3} (\sin (\frac{3 x}{2}) \frac{2}{3 x})$

Step 5.2

Combine $\sin (\frac{3 x}{2})$ and $\frac{2}{3 x}$ .

$\frac{1}{3} \cdot \frac{\sin (\frac{3 x}{2}) \cdot 2}{3 x}$

Step 5.3

Move $2$ to the left of $\sin (\frac{3 x}{2})$ .

$\frac{1}{3} \cdot \frac{2 \cdot \sin (\frac{3 x}{2})}{3 x}$

Step 5.4

Multiply $\frac{1}{3} \cdot \frac{2 \sin (\frac{3 x}{2})}{3 x}$ .

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Step 5.4.1

Multiply $\frac{1}{3}$ by $\frac{2 \sin (\frac{3 x}{2})}{3 x}$ .

$\frac{2 \sin (\frac{3 x}{2})}{3 (3 x)}$

Step 5.4.2

Multiply $3$ by $3$ .

$\frac{2 \sin (\frac{3 x}{2})}{9 x}$