<s>[INST] How to compute such a limit? Knowing $f(x,y) = 2x^2 +3y^2 -7x +15y$,  one simply proves $$|f(x,y)|\leq 5(x^2+y^2)+22 \sqrt{x^2 + y^2}$$ How can I use this info to compute
$$ \lim_{(x,y)\to(0,0)} \frac{f(x,y) - 2(x^2+y^2)^{1/4}}{(x^2+y^2)^{1/4}}\;\;\; ?$$
Thanks!
 [/INST] This can be done quite easily if you convert to polar coordinates.
We convert 
$$lim_{x, y \to (0, 0)} \frac{2x^2 + 3y^2 - 7x + 15y - 2(x^2 + y^2)^{\frac{1}{4}}}{(x^2 + y^2)^{\frac{1}{4}}}$$
turns into 
$$lim_{r \to 0} \>\> \frac{2r^2\cos^2 \theta + 3r^2 \sin^2 \theta - 7r \cos \theta + 15 r \sin \theta - 2 \sqrt{r}}{\sqrt{r}}$$
$$lim_{r \to 0} \> \>2 r^{\frac{3}{2}}\cos^2 \theta + 3 r^{\frac{3}{2}} \sin^2 \theta -7 \sqrt{r} \cos \theta + 15 \sqrt{r} \sin \theta - 2 $$
$$= -2$$
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