<s>[INST] Finding limit of multivariable function. I have a question that might be silly but I need to understand these kind of problems. So i have this limit:$$\lim_{(x,y)\to(0,0)} \frac{x^3 + y^3}{x^2 + y^2}$$
To solve it I am going to use poolar coordinates, so the limit would be like this:
$$\lim_{r\to0} \frac{r^3\cos^3\theta + r^3\sin^3\theta}{r^2}=$$
$$\lim_{r\to0} {r(\cos^3\theta + \sin^3\theta)}$$
Now it is clear to me that r tends to $r\to0$ but, can I actually say something about $\cos^3\theta + \sin^3\theta?$
In case I could, then I would say that it is bounded, thus the limit would be 0. But it is not clear to me, could anyone help me with this? 
 [/INST] It works because
$$\left|\cos^3\theta + \sin^3\theta\right| \le \left|\cos\theta\right|^3 + \left|\sin\theta\right|^3 \le 2$$
Also without polar coordinates:
\begin{align}
\left|\frac{x^3+y^3}{x^2+y^2}\right| &= \frac{|x+y||x^2-xy+y^2|}{x^2+y^2} \\
&\le|x+y| \cdot \frac{x^2+y^2+|xy|}{x^2+y^2} \\
&\le \sqrt{2}(x^2+y^2) \cdot \frac{\frac32(x^2+y^2)}{x^2+y^2}\\
&= \frac{3\sqrt{2}}2\|(x,y)\|^2 \xrightarrow{(x,y) \to (0,0)} 0
\end{align}
</s>