Calculus Examples

$\frac{lim_{x \to 0} \sin (3 x) \sin (5 x)}{x^{2}}$

Step 1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} \sin (3 x) \cdot lim_{x \to 0} \sin (5 x)}{x^{2}}$

Step 2

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

$\frac{\sin (lim_{x \to 0} 3 x) \cdot lim_{x \to 0} \sin (5 x)}{x^{2}}$

Step 3

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

$\frac{\sin (3 lim_{x \to 0} x) \cdot lim_{x \to 0} \sin (5 x)}{x^{2}}$

Step 4

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

$\frac{\sin (3 lim_{x \to 0} x) \cdot \sin (lim_{x \to 0} 5 x)}{x^{2}}$

Step 5

Move the term $5$ outside of the limit because it is constant with respect to $x$ .

$\frac{\sin (3 lim_{x \to 0} x) \cdot \sin (5 lim_{x \to 0} x)}{x^{2}}$

Step 6

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (3 \cdot 0) \cdot \sin (5 lim_{x \to 0} x)}{x^{2}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{\sin (3 \cdot 0) \cdot \sin (5 \cdot 0)}{x^{2}}$

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $3$ by $0$ .

$\frac{\sin (0) \sin (5 \cdot 0)}{x^{2}}$

Step 7.1.2

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

$\frac{0 \sin (5 \cdot 0)}{x^{2}}$

Step 7.1.3

Multiply $5$ by $0$ .

$\frac{0 \sin (0)}{x^{2}}$

Step 7.1.4

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

$\frac{0 \cdot 0}{x^{2}}$

Step 7.2

Multiply $0$ by $0$ .

$\frac{0}{x^{2}}$

Step 7.3

Divide $0$ by $x^{2}$ .

$0$