## Task Condition

Based on the definition of the derivative, find $f^{\prime}(0)$ :

$$
f(x)=\left\{\begin{array}{c}
x^{2} \cos ^{2} \frac{11}{x}, x \neq 0 \\
0, x=0
\end{array}\right.
$$
## Solution

By definition, the derivative at the point $x=0$:

$f^{\prime}(0)=\lim _{\Delta x \rightarrow 0} \frac{f(0+\Delta x)-f(0)}{\Delta x}$

Based on the definition, we find:

$$
\begin{aligned}
& f^{\prime}(0)=\lim _{\Delta x \rightarrow 0} \frac{f(0+\Delta x)-f(0)}{\Delta x}=\lim _{\Delta x \rightarrow 0} \frac{\Delta x^{2} \cos ^{2} \frac{11}{\Delta x}-0}{\Delta x}= \\
& =\lim _{\Delta x \rightarrow 0} \frac{\Delta x^{2} \cos ^{2} \frac{11}{\Delta x}}{\Delta x}=\lim _{\Delta x \rightarrow 0} \Delta x \cdot \cos ^{2} \frac{11}{\Delta x}=
\end{aligned}
$$

Since $\cos ^{2} \frac{11}{\Delta x}$ is bounded, then

$\Delta x \cdot \cos ^{2} \frac{11}{\Delta x} \rightarrow 0$, as $\Delta x \rightarrow 0$

Thus:

$=0$

Therefore, $f^{\prime}(0)=0$

## Problem Kuznetsov Differentiation $2-15$