Is this proof of limit of $f(x,y)$ correct? The limit is this: 
$$\lim\limits_{(x,y)\to(0,0)}\dfrac{\log\left(\dfrac{1}{\sqrt{x^2+y^2}}\right)}{\dfrac{1}{\sqrt{x^2+y^2}}}.$$
I think I did the proof well. Using that
$\lim\limits_{z\to\infty}\dfrac{\log(z)}{z}=0$
$,\lim\limits_{t\to 0^+}\dfrac{1}{t}=\infty$
$,\lim\limits_{(x,y)\to(0,0)}\sqrt{x^2+y^2}=0,$
$ \sqrt{x^2+y^2}>0 \space$ and the characterization of limits with sequences, we obtain that $\lim\limits_{(x,y)\to(0,0)}\dfrac{\log\left(\dfrac{1}{\sqrt{x^2+y^2}}\right)}{\dfrac{1}{\sqrt{x^2+y^2}}}=0$. 

Yes that's a fine proof, as an alternative since $\log x \le \sqrt x$ we have
$$\dfrac{\log\left(\dfrac{1}{\sqrt{x^2+y^2}}\right)}{\dfrac{1}{\sqrt{x^2+y^2}}} \le \dfrac{\sqrt{x^2+y^2}}{\sqrt[4]{x^2+y^2}}=\sqrt[4]{x^2+y^2} \to 0$$
and we can conclude by squeeze theorem.
