Calculus Examples

$lim_{x \to \frac{1}{2} \cdot π} (x - \frac{1}{2} \cdot π) \tan (3 x)$

Step 1

Simplify terms.

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Step 1.1

Simplify the limit argument.

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Step 1.1.1

Combine $π$ and $\frac{1}{2}$ .

$lim_{x \to \frac{1}{2} \cdot π} (x - \frac{π}{2}) \tan (3 x)$

Step 1.1.2

Combine terms.

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Step 1.1.2.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$lim_{x \to \frac{1}{2} \cdot π} (x \cdot \frac{2}{2} - \frac{π}{2}) \tan (3 x)$

Step 1.1.2.2

Combine $x$ and $\frac{2}{2}$ .

$lim_{x \to \frac{1}{2} \cdot π} (\frac{x \cdot 2}{2} - \frac{π}{2}) \tan (3 x)$

Step 1.1.2.3

Combine the numerators over the common denominator.

$lim_{x \to \frac{1}{2} \cdot π} \frac{x \cdot 2 - π}{2} \tan (3 x)$

Step 1.2

Combine $\frac{x \cdot 2 - π}{2}$ and $\tan (3 x)$ .

$lim_{x \to \frac{1}{2} \cdot π} \frac{(x \cdot 2 - π) \tan (3 x)}{2}$

Step 2

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

$\frac{1}{2} lim_{x \to \frac{1}{2} \cdot π} (x \cdot 2 - π) \tan (3 x)$

Step 3

Consider the left sided limit.

$lim_{x \to {(\frac{1}{2} \cdot π)}^{-}} (x \cdot 2 - π) \tan (3 x)$

Step 4

Make a table to show the behavior of the function $(x \cdot 2 - π) \tan (3 x)$ as $x$ approaches $\frac{1}{2} \cdot π$ from the left.

$\begin{array}{c} x & (x \cdot 2 - π) \tan (3 x) \\ 1.47079632 & - 0.64654562 \\ 1.56079632 & - 0.66646665 \\ 1.56979632 & - 0.66666466 \\ 1.57069632 & - 0.66666664 \end{array}$

Step 5

As the $x$ values approach $\frac{1}{2} \cdot π$ , the function values approach $- 0.667$ . Thus, the limit of $(x \cdot 2 - π) \tan (3 x)$ as $x$ approaches $\frac{1}{2} \cdot π$ from the left is $- 0.667$ .

$- 0.667$

Step 6

Consider the right sided limit.

$lim_{x \to {(\frac{1}{2} \cdot π)}^{+}} (x \cdot 2 - π) \tan (3 x)$

Step 7

Make a table to show the behavior of the function $(x \cdot 2 - π) \tan (3 x)$ as $x$ approaches $\frac{1}{2} \cdot π$ from the right.

$\begin{array}{c} x & (x \cdot 2 - π) \tan (3 x) \\ 1.67079632 & - 0.64654562 \\ 1.58079632 & - 0.66646665 \\ 1.57179632 & - 0.66666466 \\ 1.57089632 & - 0.66666664 \end{array}$

Step 8

$\frac{1}{2} \cdot - 0.667$

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{1}{2}$ and $- 0.667$ .

$\frac{- 0.667}{2}$

Step 9.2

Divide $- 0.667$ by $2$ .

$- 0.3335$