Here, $\sin\left(\frac{1}{x^{2}+1}\right)$ is continuous and definitely differentiable.

$f(x)=\sin\left(\frac{1}{x^{2}+1}\right)\\ \Rightarrow f'(x)= \frac{-2x}{\left(x^{2}+1\right)^2}\cos\left(\frac{1}{x^{2}+1}\right)\\ \Rightarrow f'(0)=0$

So the correct answer is **C**.