## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow a}\left(\frac{\sin x}{\sin a}\right)^{\frac{1}{x-a}}$
## Solution

$$
\begin{aligned}
& \lim _{x \rightarrow a}\left(\frac{\sin x}{\sin a}\right)^{\frac{1}{x-a}}=\lim _{x \rightarrow a}\left(e^{\ln \left(\frac{\sin x}{\sin a}\right)}\right)^{\frac{1}{x-a}}= \\
& =\lim _{x \rightarrow a} e^{\frac{1}{x-a} \cdot \ln \left(\frac{\sin x}{\sin a}\right)}=\exp \left\{\lim _{x \rightarrow a} \frac{1}{x-a} \cdot \ln \left(\frac{\sin x}{\sin a}\right)\right\}=
\end{aligned}
$$

Substitution:

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

$x \rightarrow a \Rightarrow y \rightarrow 0$

We get:

$$
\begin{aligned}
& =\exp \left\{\lim _{y \rightarrow 0} \frac{1}{(y+a)-a} \cdot \ln \left(\frac{\sin (y+a)}{\sin a}\right)\right\}= \\
& =\exp \left\{\lim _{y \rightarrow 0} \frac{1}{y} \cdot \ln \left(\frac{\sin y \cdot \cos a+\cos y \cdot \sin a}{\sin a}\right)\right\}= \\
& =\exp \left\{\lim _{y \rightarrow 0} \frac{1}{y} \cdot \ln (\sin y \cdot \cot a+\cos y)\right\}= \\
& =\exp \left\{\lim _{y \rightarrow 0} \frac{1}{y} \cdot \ln \left(2 \sin \frac{y}{2} \cdot \cos \frac{y}{2} \cdot \cot a+1-2 \sin ^{2} \frac{y}{2}\right)\right\}= \\
& =\exp \left\{\lim _{y \rightarrow 0} \frac{1}{y} \cdot \ln \left(1+2 \sin \frac{y}{2}\left(\cos \frac{y}{2} \cdot \cot a-\sin \frac{y}{2}\right)\right)\right\}=
\end{aligned}
$$

Using the substitution of equivalent infinitesimals:

$$
\begin{aligned}
& \ln \left(1+2 \sin \frac{y}{2}\left(\cos \frac{y}{2} \cdot \cot a-\sin \frac{y}{2}\right)\right) \sim 2 \sin \frac{y}{2}\left(\cos \frac{y}{2} \cdot \cot a-\sin \frac{y}{2}\right) \\
& y \rightarrow 0\left(2 \sin \frac{y}{2}\left(\cos \frac{y}{2} \cdot \cot a-\sin \frac{y}{2}\right) \rightarrow 0\right)
\end{aligned}
$$

We get:

$$
=\exp \left\{\lim _{y \rightarrow 0} \frac{1}{y} \cdot 2 \sin \frac{y}{2}\left(\cos \frac{y}{2} \cdot \cot a-\sin \frac{y}{2}\right)\right\}=
$$

Using the substitution of equivalent infinitesimals:

$$
\sin \frac{y}{2} \sim \frac{y}{2}_{\text {as }} y \rightarrow 0\left(\frac{y}{2} \rightarrow 0\right)
$$

We get:

$=\exp \left\{\lim _{y \rightarrow 0} \frac{1}{y} \cdot 2 \cdot \frac{y}{2} \cdot\left(\cos \frac{y}{2} \cdot \cot a-\sin \frac{y}{2}\right)\right\}=$

$=\exp \left\{\lim _{y \rightarrow 0}\left(\cos \frac{y}{2} \cdot \cot a-\sin \frac{y}{2}\right)\right\}=$

$=\exp \left\{\cos \frac{0}{2} \cdot \cot a-\sin \frac{0}{2}\right\}=\exp \{1 \cdot \cot a-0\}=$

$=\exp \{\cot a\}=e^{\cot a}$

## Problem Kuznetsov Limits 19-2