Using Maclaurin series with solving a multi-variable limits I need to determine wheter there's a limit where $(x,y)=(0,0)$ of the next function:
$$\lim_{(x,y)\to(0,0)}\frac{e^{x(y+1)}-x-1}{\sqrt{x^2+y^2}}$$
In order to simplify the expression can I use maclaurin series on $e^{x(y+1)}?$ If so it's equal to $e^{x(y+1)}=1+x(y+1)+o(x(y+1))$ and putting it on the expression will simplify it to:$\lim_{(x,y)\to(0,0)}\frac{xy+o(x(y+1))}{\sqrt{x^2+y^2}}$ and using polar I can prove the limit is equal to $0$.
Is the above correct? Can I use maclaurin series?

Note that $xy + o(x(y+1)) = xy + o(xy) + o(x) = o(x)$ so it suffices to show $\frac{o(x)}{\sqrt{x^2+y^2}} \to 0$ as $(x,y)\to(0,0)$. This is immediate since $|x| = \sqrt{x^2} \leq \sqrt{x^2+y^2}$. (That is, $o(x) \subseteq o(\sqrt{x^2+y^2})$.)
