<s>[INST] Finding $\lim_{(x,y) \to (0,0)} \frac{x^5y^5}{|x|^9 + |y|^{11}}$ Setting $y = mx^k$ suggests the limit evaluates to $0$, and as far as the techniques I know, dismisses the possibility of its non-existence. But I'm having trouble using the Squeeze Theorem to prove that it does exist. 
$$\lim_{(x,y) \to (0,0)} \frac{x^5y^5}{|x|^9 + |y|^{11}}$$
Major Edit: $(x,y) \to (0,0)$ not $(x,y) \to \infty$
 [/INST] By weighted AM-GM inequality, we get
\begin{align*}
\lvert x \rvert^9 + \lvert y \rvert^{11}
&= \frac{11}{20} \cdot \frac{20}{11} \lvert x \rvert^9 + \frac{9}{20} \cdot \frac{20}{9} \lvert y \rvert^{11} \\
&\geq \left(\frac{20}{11} \lvert x \rvert^9\right)^{11/20} \left(\frac{20}{9} \lvert y \rvert^{11}\right)^{9/20} \\
&= \text{const} \cdot \lvert x \rvert^{99/20} \lvert y \rvert^{99/20}.
\end{align*}
Hence
\begin{align*}
\frac{\lvert x \rvert^5 \lvert y \rvert^5}{\lvert x \rvert^9 + \lvert y \rvert^{11}}
\leq \text{const} \cdot \lvert x \rvert^{1/20} \lvert y \rvert^{1/20},
\end{align*}
which shows that
$$
\lim_{(x, y) \to (0, 0)} \frac{x^5 y^5}{\lvert x \rvert^9 + \lvert y \rvert^{11}} = 0.
$$
</s>