Calculus Examples

$lim_{x \to 5} \frac{\sin (3 x)}{2 x}$

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 5} \frac{\sin (3 x)}{x}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $5$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 5} \sin (3 x)}{lim_{x \to 5} x}$

Step 3

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

$\frac{1}{2} \cdot \frac{\sin (lim_{x \to 5} 3 x)}{lim_{x \to 5} x}$

Step 4

Move the term $3$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{\sin (3 lim_{x \to 5} x)}{lim_{x \to 5} x}$

Step 5

Evaluate the limits by plugging in $5$ for all occurrences of $x$ .

Tap for more steps...

Step 5.1

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{1}{2} \cdot \frac{\sin (3 \cdot 5)}{lim_{x \to 5} x}$

Step 5.2

Evaluate the limit of $x$ by plugging in $5$ for $x$ .

$\frac{1}{2} \cdot \frac{\sin (3 \cdot 5)}{5}$

Step 6

Simplify the answer.

Tap for more steps...

Step 6.1

Simplify the numerator.

Tap for more steps...

Step 6.1.1

Multiply $3$ by $5$ .

$\frac{1}{2} \cdot \frac{\sin (15)}{5}$

Step 6.1.2

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

Tap for more steps...

Step 6.1.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{1}{2} \cdot \frac{\sin (45 - 30)}{5}$

Step 6.1.2.2

Separate negation.

$\frac{1}{2} \cdot \frac{\sin (45 - (30))}{5}$

Step 6.1.2.3

Apply the difference of angles identity.

$\frac{1}{2} \cdot \frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{5}$

Step 6.1.2.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{5}$

Step 6.1.2.5

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{5}$

Step 6.1.2.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{5}$

Step 6.1.2.7

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{5}$

Step 6.1.2.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 6.1.2.8.1

Simplify each term.

Tap for more steps...

Step 6.1.2.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

Tap for more steps...

Step 6.1.2.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{5}$

Step 6.1.2.8.1.1.2

Combine using the product rule for radicals.

$\frac{1}{2} \cdot \frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{5}$

Step 6.1.2.8.1.1.3

Multiply $2$ by $3$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{5}$

Step 6.1.2.8.1.1.4

Multiply $2$ by $2$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{5}$

Step 6.1.2.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 6.1.2.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{5}$

Step 6.1.2.8.1.2.2

Multiply $2$ by $2$ .

$\frac{1}{2} \cdot \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{5}$

Step 6.1.2.8.2

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{5}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{2} (\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{5})$

Step 6.3

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{5}$ .

Tap for more steps...

Step 6.3.1

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{1}{5}$ .

$\frac{1}{2} \cdot \frac{\sqrt{6} - \sqrt{2}}{4 \cdot 5}$

Step 6.3.2

Multiply $4$ by $5$ .

$\frac{1}{2} \cdot \frac{\sqrt{6} - \sqrt{2}}{20}$

Step 6.4

Multiply $\frac{1}{2} \cdot \frac{\sqrt{6} - \sqrt{2}}{20}$ .

Tap for more steps...

Step 6.4.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{6} - \sqrt{2}}{20}$ .

$\frac{\sqrt{6} - \sqrt{2}}{2 \cdot 20}$

Step 6.4.2

Multiply $2$ by $20$ .

$\frac{\sqrt{6} - \sqrt{2}}{40}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{6} - \sqrt{2}}{40}$

Decimal Form:

$0.02588190 \dots$