Calculus Examples

$\frac{lim_{x \to 8} x \sin^{2} (7 x)}{\sin^{2} (11 x)}$

Step 1

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

$\frac{lim_{x \to 8} x \cdot lim_{x \to 8} \sin^{2} (7 x)}{\sin^{2} (11 x)}$

Step 2

Move the exponent $2$ from $\sin^{2} (7 x)$ outside the limit using the Limits Power Rule.

$\frac{lim_{x \to 8} x \cdot {(lim_{x \to 8} \sin (7 x))}^{2}}{\sin^{2} (11 x)}$

Step 3

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

$\frac{lim_{x \to 8} x \cdot \sin^{2} (lim_{x \to 8} 7 x)}{\sin^{2} (11 x)}$

Step 4

Move the term $7$ outside of the limit because it is constant with respect to $x$ .

$\frac{lim_{x \to 8} x \cdot \sin^{2} (7 lim_{x \to 8} x)}{\sin^{2} (11 x)}$

Step 5

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

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Step 5.1

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

$\frac{8 \cdot \sin^{2} (7 lim_{x \to 8} x)}{\sin^{2} (11 x)}$

Step 5.2

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

$\frac{8 \cdot \sin^{2} (7 \cdot 8)}{\sin^{2} (11 x)}$

Step 6

Simplify the answer.

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Step 6.1

Multiply by $1$ .

$\frac{8 \cdot (\sin^{2} (7 \cdot 8) \cdot 1)}{\sin^{2} (11 x)}$

Step 6.2

Multiply by $1$ .

$\frac{8 \cdot (\sin^{2} (7 \cdot 8) \cdot 1)}{\sin^{2} (11 x) \cdot 1}$

Step 6.3

Separate fractions.

$\frac{8 \cdot (1)}{1} \cdot \frac{\sin^{2} (7 \cdot 8)}{\sin^{2} (11 x)}$

Step 6.4

Convert from $\frac{1}{\sin^{2} (11 x)}$ to $\csc^{2} (11 x)$ .

$\frac{8 \cdot (1)}{1} (\sin^{2} (7 \cdot 8) \csc^{2} (11 x))$

Step 6.5

Multiply $8$ by $1$ .

$\frac{8}{1} (\sin^{2} (7 \cdot 8) \csc^{2} (11 x))$

Step 6.6

Divide $8$ by $1$ .

$8 (\sin^{2} (7 \cdot 8) \csc^{2} (11 x))$

Step 6.7

Multiply $7$ by $8$ .

$8 (\sin^{2} (56) \csc^{2} (11 x))$

Step 6.8

Evaluate $\sin (56)$ .

$8 ({0.82903757}^{2} \csc^{2} (11 x))$

Step 6.9

Raise $0.82903757$ to the power of $2$ .

$8 (0.68730329 \csc^{2} (11 x))$

Step 6.10

Multiply $0.68730329$ by $8$ .

$5.49842637 \csc^{2} (11 x)$