Calculus Examples

$lim_{x \to π} \frac{\tan (6 x)}{2 x}$

Step 1

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to π} \frac{\tan (6 x)}{x}$

Step 2

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

$\frac{1}{2} \cdot \frac{lim_{x \to π} \tan (6 x)}{lim_{x \to π} x}$

Step 3

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

$\frac{1}{2} \cdot \frac{\tan (lim_{x \to π} 6 x)}{lim_{x \to π} x}$

Step 4

Move the term $6$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{\tan (6 lim_{x \to π} x)}{lim_{x \to π} x}$

Step 5

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

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Step 5.1

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

$\frac{1}{2} \cdot \frac{\tan (6 π)}{lim_{x \to π} x}$

Step 5.2

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

$\frac{1}{2} \cdot \frac{\tan (6 π)}{π}$

Step 6

Simplify the answer.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\frac{1}{2} \cdot \frac{\tan (0)}{π}$

Step 6.1.2

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

$\frac{1}{2} \cdot \frac{0}{π}$

Step 6.2

Divide $0$ by $π$ .

$\frac{1}{2} \cdot 0$

Step 6.3

Multiply $\frac{1}{2}$ by $0$ .

$0$