Show function is continuous at $(0,0)$

Question

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function with
$$\space f(x, y) = \begin{equation}
\begin{cases}
\dfrac{y^2\log(1+x^2y^2)}{\sqrt{x^4+y^4}}, & (x, y) \neq (0, 0)\\
0, & (x, y) = 0
\end{cases}\end{equation}$$
Show that f is continious.
I've already shown that f is continous for $(x, y) \neq (0,0)$ but I am having trouble with finding a proof for $(0, 0)$ using epsilon-delta or limits of sequences. Can anyone help?
This is how far I got with simplifying for epsilon delta
$$
\Bigg{|}\frac{y^2\log(1+x^2y^2)}{\sqrt{x^4+y^4}}\Bigg{|} \\
\Leftrightarrow \Bigg{|}\frac{y^2\log(1+x^2y^2)}{\sqrt{y^4*(\frac{x^4}{y^4}+1)}}\Bigg{|} \\
\Leftrightarrow \Bigg{|}\frac{y^2\log(1+x^2y^2)}{y^2\sqrt{(\frac{x^4}{y^4}+1)}}\Bigg{|} \\
\Leftrightarrow \Bigg{|}\frac{\log(1+x^2y^2)}{\sqrt{(\frac{x^4}{y^4}+1)}}\Bigg{|} \\
\leq \big{|}\log(1+x^2y^2)\big{|}
$$
But now I am clueless on how to connect
$$
\big{|}log(1+x^2y^2)\big{|} < \epsilon \\
$$
and
$$
\sqrt{x^2+y^2} < \delta
$$

Answer 1

Ιf you use polar coordinates you will have that $$f(r,t)=\frac{1}{\sqrt{2}}\log{(1+r^4\cos^2{t}\sin^2{t}})\to^{r \to 0}0$$

So the limit as $(x,y) \to (0,0)$ is zero.