## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 1}\left(\frac{1+\cos \pi x}{\tan^{2} \pi x}\right)^{x^{2}}$
## Solution

$\lim _{x \rightarrow 1}\left(\frac{1+\cos \pi x}{\tan^{2} \pi x}\right)^{x^{2}}=\lim _{x \rightarrow 1}\left(\frac{1+1-2 \sin ^{2}\left(\frac{\pi x}{2}\right)}{\tan^{2} \pi x}\right)^{x^{2}}=$

$$
\begin{aligned}
& =\lim _{x \rightarrow 1}\left(\frac{2-2 \sin ^{2}\left(\frac{\pi x}{2}\right)}{\tan^{2} \pi x}\right)^{x^{2}}=\lim _{x \rightarrow 1}\left(\frac{2\left(1-\sin ^{2}\left(\frac{\pi x}{2}\right)\right)}{\tan^{2} \pi x}\right)^{x^{2}}= \\
& =\lim _{x \rightarrow 1}\left(\frac{2 \cos ^{2}\left(\frac{\pi x}{2}\right)}{\tan^{2} \pi x}\right)^{x^{2}}=\left(\lim _{x \rightarrow 1} \frac{2 \cos ^{2}\left(\frac{\pi x}{2}\right)}{\tan^{2} \pi x}\right)^{\lim _{x \rightarrow 1} x^{2}}= \\
& =\left(\lim _{x \rightarrow 1} \frac{2 \cos ^{2}\left(\frac{\pi x}{2}\right)}{\tan^{2} \pi x}\right)^{1^{2}}=\lim _{x \rightarrow 1} \frac{2 \cos ^{2}\left(\frac{\pi x}{2}\right)}{\tan^{2} \pi x}=
\end{aligned}
$$

Substitution:

$x=y+1 \Rightarrow y=x-1$

$x \rightarrow 1 \Rightarrow y \rightarrow 0$

We get:

$$
\begin{aligned}
& =\lim _{y \rightarrow 0} \frac{2 \cos ^{2}\left(\frac{\pi(y+1)}{2}\right)}{\tan^{2} \pi(y+1)}=\lim _{y \rightarrow 0} \frac{2 \cos ^{2}\left(\frac{\pi y}{2}+\frac{\pi}{2}\right)}{\tan^{2}(\pi y+\pi)}= \\
& =\lim _{y \rightarrow 0} \frac{2\left(-\sin \left(\frac{\pi y}{2}\right)\right)^{2}}{\tan^{2} \pi y}=\lim _{y \rightarrow 0} \frac{2 \sin ^{2}\left(\frac{\pi y}{2}\right)}{\tan^{2} \pi y}=
\end{aligned}
$$

Using the substitution of equivalent infinitesimals:

![](https://cdn.mathpix.com/cropped/2024_05_22_08a731a792d2ab45f373g-15.jpg?height=146&width=845&top_left_y=1564&top_left_x=152)

$\tan \pi y \sim \pi y$, as $y \rightarrow 0(\pi y \rightarrow 0)$

We get:

$$
=\lim _{y \rightarrow 0} \frac{2\left(\frac{\pi y}{2}\right)^{2}}{(\pi y)^{2}}=\lim _{y \rightarrow 0} \frac{(\pi y)^{2}}{2(\pi y)^{2}}=\lim _{y \rightarrow 0} \frac{1}{2}=\frac{1}{2}
$$

## Problem Kuznetsov Limits 20-30