Calculus Examples

$lim_{x \to \frac{π}{4}} {\tan (x)}^{\tan (2 x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

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

$lim_{x \to \frac{π}{4}} e^{\ln ({\tan (x)}^{\tan (2 x)})}$

Step 1.2

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

$lim_{x \to \frac{π}{4}} e^{\tan (2 x) \ln (\tan (x))}$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to \frac{π}{4}} \tan (2 x) \ln (\tan (x))}$

Step 3

Consider the left sided limit.

$lim_{x \to {(\frac{π}{4})}^{-}} \tan (2 x) \ln (\tan (x))$

Step 4

Make a table to show the behavior of the function $\tan (2 x) \ln (\tan (x))$ as $x$ approaches $\frac{π}{4}$ from the left.

$\begin{array}{c} x & \tan (2 x) \ln (\tan (x)) \\ 0.68539816 & - 0.99327506 \\ 0.77539816 & - 0.99993332 \\ 0.78439816 & - 0.99999933 \end{array}$

Step 5

As the $x$ values approach $\frac{π}{4}$ , the function values approach $- 1$ . Thus, the limit of $\tan (2 x) \ln (\tan (x))$ as $x$ approaches $\frac{π}{4}$ from the left is $- 1$ .

$- 1$

Step 6

Consider the right sided limit.

$lim_{x \to {(\frac{π}{4})}^{+}} \tan (2 x) \ln (\tan (x))$

Step 7

Make a table to show the behavior of the function $\tan (2 x) \ln (\tan (x))$ as $x$ approaches $\frac{π}{4}$ from the right.

$\begin{array}{c} x & \tan (2 x) \ln (\tan (x)) \\ 0.88539816 & - 0.99327506 \\ 0.79539816 & - 0.99993332 \\ 0.78639816 & - 0.99999933 \end{array}$

Step 8

As the $x$ values approach $\frac{π}{4}$ , the function values approach $- 1$ . Thus, the limit of $\tan (2 x) \ln (\tan (x))$ as $x$ approaches $\frac{π}{4}$ from the right is $- 1$ .

$e^{- 1}$

Step 9

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{e}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{e}$

Decimal Form:

$0.36787944 \dots$