Calculus Examples

$lim_{x \to 0} x \tan (x + \frac{π}{2})$

Step 1

Combine terms.

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Step 1.1

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

$lim_{x \to 0} x \tan (x \cdot \frac{2}{2} + \frac{π}{2})$

Step 1.2

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

$lim_{x \to 0} x \tan (\frac{x \cdot 2}{2} + \frac{π}{2})$

Step 1.3

Combine the numerators over the common denominator.

$lim_{x \to 0} x \tan (\frac{x \cdot 2 + π}{2})$

Step 2

Consider the left sided limit.

$lim_{x \to 0^{-}} x \tan (\frac{x \cdot 2 + π}{2})$

Step 3

Make a table to show the behavior of the function $x \tan (\frac{x \cdot 2 + π}{2})$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & x \tan (\frac{x \cdot 2 + π}{2}) \\ - 0.1 & - 0.99666444 \\ - 0.01 & - 0.99996666 \\ - 0.001 & - 0.99999966 \end{array}$

Step 4

As the $x$ values approach $0$ , the function values approach $- 1$ . Thus, the limit of $x \tan (\frac{x \cdot 2 + π}{2})$ as $x$ approaches $0$ from the left is $- 1$ .

$- 1$

Step 5

Consider the right sided limit.

$lim_{x \to 0^{+}} x \tan (\frac{x \cdot 2 + π}{2})$

Step 6

Make a table to show the behavior of the function $x \tan (\frac{x \cdot 2 + π}{2})$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & x \tan (\frac{x \cdot 2 + π}{2}) \\ 0.1 & - 0.99666444 \\ 0.01 & - 0.99996666 \\ 0.001 & - 0.99999966 \end{array}$

Step 7

As the $x$ values approach $0$ , the function values approach $- 1$ . Thus, the limit of $x \tan (\frac{x \cdot 2 + π}{2})$ as $x$ approaches $0$ from the right is $- 1$ .

$- 1$