## Problem Statement

Calculate the limit of the function:
$\lim _{x \rightarrow a}\left(\frac{\sin x-\sin a}{x-a}\right)^{\frac{x^{2}}{a^{2}}}$
## Solution

$\lim _{x \rightarrow a}\left(\frac{\sin x-\sin a}{x-a}\right)^{\frac{x^{2}}{a^{2}}}=\left(\lim _{x \rightarrow a} \frac{\sin x-\sin a}{x-a}\right)^{\lim _{x \rightarrow a} \frac{x^{2}}{a^{2}}}=$

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

Using the substitution of equivalent infinitesimals:

$$
\sin \frac{x-a}{2} \sim \frac{x-a}{2}, \text { as } \quad x \rightarrow a\left(\frac{x-a}{2} \rightarrow 0\right)
$$

We get:

$$
=\lim _{x \rightarrow a} \frac{2 \cdot \frac{x-a}{2} \cdot \cos \frac{x+a}{2}}{x-a}=\lim _{x \rightarrow a} \cos \frac{x+a}{2}=\cos \frac{a+a}{2}=\cos a
$$

## Problem Kuznetsov Limits 20-15