## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0}\left(\tan\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}$
## Solution

$$
\begin{aligned}
& \lim _{x \rightarrow 0}\left(\tan\left(\frac{\pi}{4}-x\right)\right)^{\left(e^{x}-1\right) / x}=\left(\lim _{x \rightarrow 0} \tan\left(\frac{\pi}{4}-x\right)\right)^{\lim _{x \rightarrow 0}\left(e^{x}-1\right) / x}= \\
& =\left(\tan\left(\frac{\pi}{4}-0\right)\right)^{\lim _{x \rightarrow 0}\left(e^{x}-1\right) / x}=(1)^{\lim _{x \rightarrow 0}\left(e^{x}-1\right) / x}=
\end{aligned}
$$

Using the substitution of equivalent infinitesimals:

$e^{x}-1 \sim x$, as $x \rightarrow 0$

We get:

$=1^{\lim _{x \rightarrow 0} \frac{x}{x}}=1^{\lim _{x \rightarrow 0} 1}=1^{1}=1$

## Problem Kuznetsov Limits 18-27