Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

Set up the limit as a left-sided limit.

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

Step 3

Evaluate the limits by plugging in the value for the variable.

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Step 3.1

Evaluate the limit of $e^{\tan (\frac{2 x}{π}) \ln (x - 2)}$ by plugging in $1$ for $x$ .

$e^{\tan (\frac{2 \cdot 1}{π}) \ln (1 - 2)}$

Step 3.2

Since $e^{\tan (\frac{2 \cdot 1}{π}) \ln (1 - 2)}$ is undefined, the limit does not exist.

$Does not exist$

Step 4

Set up the limit as a right-sided limit.

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

Step 5

Evaluate the limits by plugging in the value for the variable.

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Step 5.1

Evaluate the limit of $e^{\tan (\frac{2 x}{π}) \ln (x - 2)}$ by plugging in $1$ for $x$ .

$e^{\tan (\frac{2 \cdot 1}{π}) \ln (1 - 2)}$

Step 5.2

Since $e^{\tan (\frac{2 \cdot 1}{π}) \ln (1 - 2)}$ is undefined, the limit does not exist.

$Does not exist$

Step 6

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$