Calculus Examples

$lim_{x \to 8} \frac{\sin^{2} (x)}{3 x^{2}}$

Step 1

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

$\frac{1}{3} lim_{x \to 8} \frac{\sin^{2} (x)}{x^{2}}$

Step 2

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $8$ .

$\frac{1}{3} \cdot \frac{lim_{x \to 8} \sin^{2} (x)}{lim_{x \to 8} x^{2}}$

Step 3

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

$\frac{1}{3} \cdot \frac{{(lim_{x \to 8} \sin (x))}^{2}}{lim_{x \to 8} x^{2}}$

Step 4

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

$\frac{1}{3} \cdot \frac{\sin^{2} (lim_{x \to 8} x)}{lim_{x \to 8} x^{2}}$

Step 5

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

$\frac{1}{3} \cdot \frac{\sin^{2} (lim_{x \to 8} x)}{{(lim_{x \to 8} x)}^{2}}$

Step 6

Evaluate the limits by plugging in $8$ for all occurrences of $x$ .

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Step 6.1

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{1}{3} \cdot \frac{\sin^{2} (8)}{{(lim_{x \to 8} x)}^{2}}$

Step 6.2

Evaluate the limit of $x$ by plugging in $8$ for $x$ .

$\frac{1}{3} \cdot \frac{\sin^{2} (8)}{8^{2}}$

Step 7

Simplify the answer.

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Step 7.1

Combine.

$\frac{1 \sin^{2} (8)}{3 \cdot 8^{2}}$

Step 7.2

Multiply $\sin^{2} (8)$ by $1$ .

$\frac{\sin^{2} (8)}{3 \cdot 8^{2}}$

Step 7.3

Raise $8$ to the power of $2$ .

$\frac{\sin^{2} (8)}{3 \cdot 64}$

Step 7.4

Simplify the numerator.

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Step 7.4.1

Evaluate $\sin (8)$ .

$\frac{{0.1391731}^{2}}{3 \cdot 64}$

Step 7.4.2

Raise $0.1391731$ to the power of $2$ .

$\frac{0.01936915}{3 \cdot 64}$

Step 7.5

Multiply $3$ by $64$ .

$\frac{0.01936915}{192}$

Step 7.6

Divide $0.01936915$ by $192$ .

$0.00010088$