Calculus Examples

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

Step 1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $3$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

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

Step 5

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

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

Step 6

Evaluate the limits by plugging in $3$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 6.2

Evaluate the limit of $x$ by plugging in $3$ for $x$ .

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

Step 7

Simplify the answer.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Multiply $2$ by $3$ .

$\frac{\sin^{2} (6)}{3^{2}}$

Step 7.1.2

Evaluate $\sin (6)$ .

$\frac{{0.10452846}^{2}}{3^{2}}$

Step 7.1.3

Raise $0.10452846$ to the power of $2$ .

$\frac{0.01092619}{3^{2}}$

Step 7.2

Raise $3$ to the power of $2$ .

$\frac{0.01092619}{9}$

Step 7.3

Divide $0.01092619$ by $9$ .

$0.00121402$