<s>[INST] Find an upperbound for the rational function I know
$$\lim_{(x, y)\to (0,0)} \frac{x^3 + y^4}{x^2 + y^2} = 0$$ so 
$$\left\lvert \frac{x^3 + y^4}{x^2  + y^2} \right\rvert \le f(x, y)$$ for some simpler $f(x, y)$  whose limit is also $0$.
How do I find the function $f(x, y)$? In other words, how do I get the upper bound?
 [/INST] Using the inequalities $x^2+y^2\ge x^2$ and $x^2+y^2\ge y^2$, we can write
$$\begin{align}
\left|\frac{x^3+y^4}{x^2+y^2}\right|&\le \frac{|x|^3}{x^2+y^2}+\frac{y^4}{x^2+y^2}\\\\
&\le |x|+y^2
\end{align}$$
</s>