Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Move the exponent $3$ from $x^{3}$ outside the limit using the Limits Power Rule.

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

Step 4

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

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

Step 5

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

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

Step 6

Evaluate the limit of $1$ which is constant as $x$ approaches $8$ .

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

Step 7

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

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

Step 8

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

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Step 8.1

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

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

Step 8.2

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

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

Step 9

Simplify the answer.

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Step 9.1

Raise $8$ to the power of $3$ .

$\sin (512) \cdot \sin (\frac{1}{8^{2}})$

Step 9.2

Remove full rotations of $360$ ° until the angle is between $0$ ° and $360$ °.

$\sin (152) \cdot \sin (\frac{1}{8^{2}})$

Step 9.3

Evaluate $\sin (152)$ .

$0.46947156 \cdot \sin (\frac{1}{8^{2}})$

Step 9.4

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

$0.46947156 \cdot \sin (\frac{1}{64})$

Step 9.5

Evaluate $\sin (\frac{1}{64})$ .

$0.46947156 \cdot 0.0002727$

Step 9.6

Multiply $0.46947156$ by $0.0002727$ .

$0.00012802$