Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(2 - x)}^{\tan (\frac{π x}{2})}$ as $e^{\ln ({(2 - x)}^{\tan (\frac{π x}{2})})}$ .

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

Step 1.2

Expand $\ln ({(2 - x)}^{\tan (\frac{π x}{2})})$ by moving $\tan (\frac{π x}{2})$ outside the logarithm.

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

Step 2

Move the limit into the exponent.

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

Step 3

Consider the left sided limit.

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

Step 4

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

$\begin{array}{c} x & \tan (\frac{π x}{2}) \ln (2 - x) \\ 0.9 & 0.60176479 \\ 0.99 & 0.63340563 \\ 0.999 & 0.63630115 \\ 0.9999 & 0.63658793 \\ 0.99999 & 0.63661658 \end{array}$

Step 5

As the $x$ values approach $1$ , the function values approach $0.637$ . Thus, the limit of $\tan (\frac{π x}{2}) \ln (2 - x)$ as $x$ approaches $1$ from the left is $0.637$ .

$0.637$

Step 6

Consider the right sided limit.

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

Step 7

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

$\begin{array}{c} x & \tan (\frac{π x}{2}) \ln (2 - x) \\ 1.1 & 0.66522011 \\ 1.01 & 0.63977162 \\ 1.001 & 0.63693777 \\ 1.0001 & 0.6366516 \end{array}$

Step 8

As the $x$ values approach $1$ , the function values approach $0.637$ . Thus, the limit of $\tan (\frac{π x}{2}) \ln (2 - x)$ as $x$ approaches $1$ from the right is $0.637$ .

$e^{0.637}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$e^{0.637}$

Decimal Form:

$1.89079996 \dots$