# Multivariable Limit problem , epsilon-delta

I am trying to solve this delta-epsilon problem, but I did not find an effective way to find the following limit:

$$\lim_{(x,y) \to (-3, 4)} \frac{2x^3 + 5y^3 + 18x^2 + 54x - 60y^2 + 240y - 266}{\sqrt{x^2 + 6x + 25 + y^2 - 8y}}$$

I actually tried a lot of inequalities (such as Cauchy-Schwarz), but nothing came up with this demonstration. How can I solve this problem?

• make the substitution u=x+3, v=y-4, and switch into polar coordinatws – Saketh Malyala Jun 28 '17 at 8:35

After letting $x:=-3+r\cos(t)$ and $y:=4+r\sin(t)$ the limit becomes $$\lim_{r\to 0}\frac{r^3(2\cos^3 t+5\sin^3 t)}{r}=\lim_{r\to 0}r^2(2\cos^3 t+5\sin^3 t)=0$$ (note that $|(2\cos^3 t+5\sin^3 t)|\leq 2+5=7$).