Proving that $\lim\limits_{(x, y)\to (0, 0)} \frac{x^3 y^3}{(x^4+y^6)\sqrt{x^2+y^2}}$ doesn't exist I want to prove that $\displaystyle \lim_{(x, y)\to (0, 0)} \frac{x^3 y^3}{(x^4+y^6)\sqrt{x^2+y^2}}$ doesn't exist
I denoted $f(x, y):=\frac{x^3 y^3}{(x^4+y^6)\sqrt{x^2+y^2}}$. I tried looking at $f(\frac{1}{n}, \frac{1}{n})$ and other similar sequences, but no result. No matter what sequence I chose I got that the limit is $0$. However, WA says that this doesn't exist. What sequence should I pick?

As Maxim pointed out the limit actually exists and is equal to 0. I give an alternative proof for that now. It suffices to show, that for |x|,|y| sufficiently small there holds
$$\left| \frac{x^3y^3}{(x^4+y^6)\sqrt{x^2+y^2}}\right| \le \sqrt{2}\max(\sqrt{|y|},|x|^{\frac{1}{3}}).$$
This we see at follows. First by Cauchy-Schwart and the triangle inequality there holds
$$\frac{1}{\sqrt{2}}(|x|+|y|)\le \sqrt{x^2+y^2}\le |x|+|y|.$$
We assume in the following w.o.l.g. that $1>>x,y>0$. Then,
$$\frac{x^3y^3}{(x^4+y^6)\sqrt{x^2+y^2}}\le \sqrt{2} \frac{x^3y^3}{(x^4+y^6)(x+y)}.$$
First assume, that $x\ge y$. Then
$$\frac{x^3y^3}{(x^4+y^6)\sqrt{x^2+y^2}}\le \sqrt{2} \frac{x^3y^3}{(x^4+y^6)(x+y)}\le \sqrt{2}\frac{x^2y^3}{x^4} =\le \sqrt{2} \frac{y^3}{x^2}\le \sqrt{2} x$$
and the above statement is fulfilled. We now assume in the following, that $x\le y$. Then
$$\frac{x^3y^3}{(x^4+y^6)\sqrt{x^2+y^2}}\le \sqrt{2} \frac{x^3y^2}{x^4+y^6}.$$
Now we consider the two cases $x^4\le y^6$ and $x^4\ge y^6$. In the first case,
$$\frac{x^3y^2}{x^4+y^6} \le \frac{x^3y^2}{y^6}\le \frac{x^3}{y^4} \le \frac{y^\frac{3*6}{4}}{y^4} =  \sqrt{y}.$$
In the remaining case,
$$\frac{x^3y^2}{x^4+y^6} \le \frac{x^3y^2}{x^4} = \frac{y^2}{x}\le x^\frac{1}{3}.$$
Therefore, the statement is proved.
The following reasoning does not yield the statement, as Maxim pointed out:
Set $y=y(x)=x^q$ with $q>0$. You can try to tune $q$, such that the numerator dominates the denominator. By the above, $q$ has to be smaller than 1. For the denominator there holds
$$(x^4+x^{6q}) \sqrt{x^2+x^{2q}} = x^{\min(4,6q)}(1+O(1)) x^{min(1,q)} (1+O(1))$$
for $x\to0$. E.g., for $q=4/6$ we have
$$x^{3+3q} = x^5 > x^{28/6}(1+o(1)) =  x^{7q}(1+o(1)).$$
Thus for that choice, the sequence will diverge (This last deduction is wrong).
