I'm interested in checking if the following function $$ f(x,y) = \begin{cases} \dfrac{\sin(x+y)}{\sqrt{x^2+y^2}},& \text{if } (x,y) \neq (0,0),\\ 0, &\text{if } (x,y) = (0,0). \end{cases} $$ is differentiable at the origin, and also I would like to compute its partial derivatives at the origin, if possible. For the differentiability, my claim is that $f$ is not differentiable at $(0,0)$ since it's not continuous at $(0,0)$. Indeed, if we compute the limit in polar coordinates, we obtain that $$ \lim_{r \to 0} \dfrac{\sin(r(\cos(\theta)+ \sin(\theta)))}{r} = \begin{cases} \cos(\theta) + \sin(\theta),& \text{if } \cos(\theta) + \sin(\theta) \neq 0,\\ 0, &\text{otherwise}. \end{cases} $$ therefore the limit $$ \lim_{(x,y) \to (0,0)} \dfrac{\sin(x+y)}{\sqrt{x^2 + y^2}} $$ doesn't exist. For the partial derivatives at the origin, I obtain the following $$ \dfrac{\partial f}{\partial x}(0,0) = \lim_{h \to 0} \dfrac{\sin(h)}{|h|h} = +\infty $$ and the same for the partial derivative with respect to $y$ at $(0,0)$. However, Wolfram Alpha says that the partial derivatives at the origin are both equal to $0$.
QUESTION: Is my reasoning right or have I done a mistake?