Single Variable Calculus, Chapter 3, 3.1, Section 3.1, Problem 51

Determine whether $f'(0)$ exists in the function
$
\displaystyle
f(x) = \left\{
\begin{array}{c}
x \sin\left(\frac{1}{x}\right) & \text{if} & x \neq 0\\
0 & \text{if} & x = 0
\end{array}\right.
$



Based from the definition,



$
\displaystyle
f'(a) = \lim\limits_{x \to a} \frac{f(x) - f(a)}{x-a}
$


$
\begin{equation}
\begin{aligned}
f'(0) & = \lim\limits_{x \to 0} \frac{x \sin \left( \frac{1}{x}\right) - f(0)}{x-0}\\
f'(0) & = \lim\limits_{x \to 0} \frac{\cancel{x}\sin \left( \frac{1}{x}\right)}{\cancel{x}}\\
f'(0) & = \lim\limits_{x \to 0} \sin \left(\frac{1}{x}\right)\\
f'(0) & = \sin \frac{1}{0}
\end{aligned}
\end{equation}
$



Thus, $f'(0)$ does not exist.