Calculus Examples

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

Step 1

Combine terms.

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

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

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

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

Combine the numerators over the common denominator.

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

Step 2

Combine factors.

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Multiply $2$ by $- 1$ .

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

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

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

Step 3

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

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

Step 4

Consider the left sided limit.

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

Step 5

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

$\begin{array}{c} x & (π - 2 x) \tan (x) \\ 1.47079632 & 1.99332888 \\ 1.56079632 & 1.99993333 \\ 1.56979632 & 1.99999933 \end{array}$

Step 6

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

$2$

Step 7

Consider the right sided limit.

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

Step 8

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

$\begin{array}{c} x & (π - 2 x) \tan (x) \\ 1.67079632 & 1.99332888 \\ 1.58079632 & 1.99993333 \\ 1.57179632 & 1.99999933 \end{array}$

Step 9

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

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

Step 10

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

$1$