Calculus Examples

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

Step 1

Combine terms.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

Combine the numerators over the common denominator.

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

Step 2

Simplify the limit argument.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Multiply $\frac{2}{x \cdot 2 - π}$ by $1$ .

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

Step 3

Consider the left sided limit.

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

Step 4

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

$\begin{array}{c} x & \tan (x) + \frac{2}{x \cdot 2 - π} \\ 1.47079632 & - 0.03335557 \\ 1.56079632 & - 0.00333335 \\ 1.56979632 & - 0.00033333 \\ 1.57069632 & - 0.00003349 \end{array}$

Step 5

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

$0$

Step 6

Consider the right sided limit.

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

Step 7

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

$\begin{array}{c} x & \tan (x) + \frac{2}{x \cdot 2 - π} \\ 1.67079632 & 0.03335557 \\ 1.58079632 & 0.00333335 \\ 1.57179632 & 0.00033333 \\ 1.57089632 & 0.00003315 \end{array}$

Step 8

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

$0$