Calculus Examples

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

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent $2$ from $\sin^{2} (3 x)$ outside the limit using the Limits Power Rule.

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply $3$ by $0$ .

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

Step 3.1.2

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

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

Step 3.1.3

Raising $0$ to any positive power yields $0$ .

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

Step 3.2

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

$0$