Step-by-step Solution:

\[ f(x) = \begin{cases} x^2 \sin \left( \frac{1}{x} \right), & x \neq 0 \\ 0, & x = 0 \end{cases} \] \[ m = \frac{dy}{dx} = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} \] \[ \lim_{x \to 0} \frac{x^2 \sin \left( \frac{1}{x} \right) - 0}{x} \] \[ \lim_{x \to 0} x \sin \left( \frac{1}{x} \right) \] \[ -1 \leq \sin \left( \frac{1}{x} \right) \leq 1 \] \[ \Rightarrow - x \leq x \sin \left( \frac{1}{x} \right) \leq x \] \[ \Rightarrow \lim_{x \to 0} (-x) \leq \lim_{x \to 0} x \sin \left( \frac{1}{x} \right) \leq \lim_{x \to 0} x \] \[ \Rightarrow 0 \leq \lim_{x \to 0} x^2 \sin \left( \frac{1}{x} \right) \leq 0 \] \[ \therefore \lim_{x \to 0} x^2 \sin \left( \frac{1}{x} \right) = 0 \]