Proof that $\frac{xy^2}{x^2+y^6}$ is unbounded on $\mathbb{R}^2$ I am trying to prove that $\frac{xy^2}{x^2+y^6}$ is unbounded. Is the following correct?
Consider arbitrary $M \in \mathbb{R} \setminus\{0\}$ (the case in which $M=0$ is trivial). Set $y:=\sqrt{\frac{|x|}{2}}$. Thus, for any $x$ $\in \mathbb{R} \setminus \{0\}$, we have that $|x|>y^2$, and so that $\frac{xy^2}{x^2+y^6}>\frac{y^4}{x^2+y^6}$. We now argue that there exists an $x \in \mathbb{R}$ such that $\frac{xy^2}{x^2+y^6}>\frac{y^4}{x^2+y^6}>M$.
Note that $\frac{y^4}{x^2+y^6}$ is monotonically increasing as $x \rightarrow 0$. Also, note that $\frac{y^4}{x^2+y^6}$ is bounded above by $\frac{1}{y^2}=\frac{2}{x}$. Thus, as $x \rightarrow 0$, $\frac{y^4}{x^2+y^6}$ becomes arbitrarily large (as the upper bound towards which it is monotonically increasing becomes arbitrarily large). Thus, there must exist an $x \in \mathbb{R}$ such that $\frac{xy^2}{x^2+y^6}>M$, for $(x,y):=(x, \sqrt{\frac{x}{2}})$.

Let $x=\frac{1}{n^3}$, $y=\frac{1}{n}$, where $n$ is a positive integer. Then
$$
\frac{xy^2}{x^2+y^6}=\frac{n}{2},
$$
which $\to \infty$ when $n \to \infty$.
