Calculus Examples

$\frac{lim_{x \to 0} \tan^{2} (3 x)}{2 x}$

Step 1

Evaluate the limit.

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Step 1.1

Move the exponent

$2$ from

$\tan^{2} (3 x)$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} \tan (3 x))}^{2}}{2 x}$

Step 1.2

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

$\frac{\tan^{2} (lim_{x \to 0} 3 x)}{2 x}$

Step 1.3

Move the term

$3$ outside of the limit because it is constant with respect to

$x$ .

$\frac{\tan^{2} (3 lim_{x \to 0} x)}{2 x}$

Step 2

Evaluate the limit of

$x$ by plugging in

$0$ for

$x$ .

$\frac{\tan^{2} (3 \cdot 0)}{2 x}$

Step 3

Simplify the answer.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Multiply

$3$ by

$0$ .

$\frac{\tan^{2} (0)}{2 x}$

Step 3.1.2

The exact value of

$\tan (0)$ is

$0$ .

$\frac{0^{2}}{2 x}$

Step 3.1.3

Raising

$0$ to any positive power yields

$0$ .

$\frac{0}{2 x}$

Step 3.2

Cancel the common factor of

$0$ and

$2$ .

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Step 3.2.1

Factor

$2$ out of

$0$ .

$\frac{2 (0)}{2 x}$

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor

$2$ out of

$2 x$ .

$\frac{2 (0)}{2 (x)}$

Step 3.2.2.2

Cancel the common factor.

$\frac{2 \cdot 0}{2 x}$

Step 3.2.2.3

Rewrite the expression.

$\frac{0}{x}$

Step 3.3

Divide

$0$ by

$x$ .

$0$