<s>[INST] On the differentiability of $\frac{\sin(x^2)+\cos(y^2)}{\sqrt{x^2+y^2}}$ Does this function $f:\mathbb{R^2}\to\mathbb{R}$ defined by:
$$\frac{\sin(x^2)+\cos(y^2)}{\sqrt{x^2+y^2}}$$
differentiable at the origin? Thank you
 [/INST] The function
$$
f(x) = \frac{\sin \left(x^2\right)+\cos \left(y^2\right)}{\sqrt{x^2+y^2}}
$$
is not differentiable at the origin.

The iterated limit demonstrates the problem.

$$
\begin{align}
 \lim_{\color{red}{x}\to 0} f(x,y) &= \frac{\cos \left(y^2\right)}{\sqrt{y^2}} \\[3pt]
 \lim_{\color{blue}{y}\to 0} \left( \lim_{\color{red}{x}\to 0} f(x,y) \right) &= \infty \\
\end{align}
$$

$$
\begin{align}
 \lim_{\color{blue}{y}\to 0} f(x,y) &= \frac{\sin \left(x^2\right)+1}{\sqrt{x^2}} \\[3pt]
 \lim_{\color{red}{x}\to 0} \left( \lim_{\color{blue}{y}\to 0} f(x,y) \right) &= \infty \\
\end{align}
$$

The function is growing rapidly near the origin.



Hint: Maclaurin series

$$  
\sin x^{2} = x^{2} - \frac{x^{2}}{6} + \mathcal{O}\left( x^{8} \right)
$$
</s>