<s>[INST] Continuity of $\frac{x^5-4x^3y^2-xy^4}{(x^2+y^2)^2}$ at (0, 0) I am having trouble proving that $\dfrac{x^5-4x^3y^2-xy^4}{(x^2+y^2)^2}$ is continuous at $(0, 0)$ if we set the value at $(0, 0)$ to be $0$.
I don't see a way to prove this as I cannot factor this into partial fractions.
 [/INST] Hint. By considering polar coordinates,
$$
x:=r\cos\theta,\quad y:=r\sin\theta,
$$ you get
$$
\begin{align}
f(r\cos\theta,r\sin\theta)&=\frac{r^5(\cos^5\theta-4\cos^3\theta \sin^2\theta-\cos\theta\sin^4\theta)}{r^4}\\\\
&=r(\cos^5\theta-4\cos^3\theta \sin^2\theta-\cos\theta\sin^4\theta)
\end{align}
$$ then observe that
$$
\left|\cos^5\theta-4\cos^3\theta \sin^2\theta-\cos\theta\sin^4\theta\right|\leq7
$$ gives

$$
\left|f(r\cos\theta,r\sin\theta)\right|\leq7r
$$ 

for all $r>0$ and $\theta$.
</s>