i have a problem with this exercise.
Given the function $ \left\{\begin{matrix} (\frac{x^2y}{x^4+y^2})^2 if (x,y)\neq (0,0) \\ 0 if (x,y)=(0,0)\end{matrix}\right. $ test its continuity.
The function is defined for $ {(x,y)\in \mathbb{R}^2:x^4+y^2\neq0} $, so it's a continuos function in its domain and it's worth zero in $ (0,0) $. To test the continuinity in point i pass to the polar coordinates. Now, i'm arrived to proof here that:
$ \lim_{\rho ->0^+}\frac{\rho^2}{\rho^4+1}=0 $
But, parameterizing with $ \gamma=\left\{\begin{matrix}x=t\\ y=t^2\end{matrix}\right. $ i've proved that $ \lim_{t->0}(\frac{t^2}{2t^4})^2=\frac{1}{4} $.
What am i doing wrong?
Thanks for any help!
From $$\rho^4\cos^8 \theta + \sin^4 \theta \le \rho^4 + 1,$$
we can't conclude that
$$\frac1{\rho^4\cos^8 \theta + \sin^4 \theta }\le \frac1{\rho^4 + 1}$$
$\frac1x$ is a decreasing function over the positive domain.
Also, $(a+b)^2 \ne a^2 + b^2$ in general.
In fact the limit doesn't exist. If you travel along $x=0$, we get $0$ but if we follow the path $y=x^2$, we get $\frac14$.