Calculus Examples

$\frac{lim_{x \to 0} \sin^{2} (7 x)}{\sin^{2} (11 x)}$

Step 1

Evaluate the limit.

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Step 1.1

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

$\frac{{(lim_{x \to 0} \sin (7 x))}^{2}}{\sin^{2} (11 x)}$

Step 1.2

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

$\frac{\sin^{2} (lim_{x \to 0} 7 x)}{\sin^{2} (11 x)}$

Step 1.3

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

$\frac{\sin^{2} (7 lim_{x \to 0} x)}{\sin^{2} (11 x)}$

Step 2

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

$\frac{\sin^{2} (7 \cdot 0)}{\sin^{2} (11 x)}$

Step 3

Simplify the answer.

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Step 3.1

Multiply by $1$ .

$\frac{\sin^{2} (7 \cdot 0)}{\sin^{2} (11 x) \cdot 1}$

Step 3.2

Separate fractions.

$\frac{\sin^{2} (7 \cdot 0)}{1} \cdot \frac{1}{\sin^{2} (11 x)}$

Step 3.3

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

$\frac{\sin^{2} (7 \cdot 0)}{1} \csc^{2} (11 x)$

Step 3.4

Multiply $7$ by $0$ .

$\frac{\sin^{2} (0)}{1} \csc^{2} (11 x)$

Step 3.5

Divide $\sin^{2} (0)$ by $1$ .

$\sin^{2} (0) \csc^{2} (11 x)$

Step 3.6

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

$0^{2} \csc^{2} (11 x)$

Step 3.7

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

$0 \csc^{2} (11 x)$

Step 3.8

Multiply $0$ by $\csc^{2} (11 x)$ .

$0$