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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?

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    $\begingroup$ make the substitution u=x+3, v=y-4, and switch into polar coordinatws $\endgroup$ – Saketh Malyala Jun 28 '17 at 8:35
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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 $|(2\cos^3 t+5\sin^3 t)|\leq 2+5=7$).

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  • $\begingroup$ thanks Robert Z, its helpful. actually, this method was not im my consideration, but now it is. $\endgroup$ – Wilfred V Jun 28 '17 at 17:08
  • $\begingroup$ @Wilfred V You are welcome. BTW if you are new here take a few minutes for a tour math.stackexchange.com/tour $\endgroup$ – Robert Z Jun 28 '17 at 17:10

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