<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] Let $x=r\cos{t}, y=r\sin{t}$, then $$\left\lvert \frac{x^3 + y^4}{x^2  + y^2} \right\rvert=|r(\cos^3{t}+r\sin^4{t})|\leq |r||1+r| = \sqrt{x^2+y^2}+x^2+y^2$$
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