The limit of xye^-(x+y)^2 when x^2+y^2 approach infinity

  • Thread starter Mahathepp
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I try to figure it out but I can't get the answer that I need and when I look upon the solution from the book I don't understand it at all. The answer is " no limit" and there is no explanation why. The question is

Determine the limit of

lim (x2+y2)- -> infinity (xye-(x+y)2

in this case I use polar coordinate which I get

lim r2 - -> infinity ( r2cos(x)sin(X) / er^2(1+sin(2x) )

My idea is since there is (er^2(1+sin(2x)) in denominator which is depening on angle (2x) but I am not sure if I understand correct. Can anyone here help me to figure it out? Thanks in advance.

Regard
 
Last edited:

Orodruin

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If the limit existed, then the result would not depend on how r approaches infinity. As you can see, you will obtain different results depending on the polar coordinate (if you fix it) and therefore the limit is not well defined.
 

mathman

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If x+y = 0, the limit is -∞. If x+y -> ∞, the limit = 0. Manipulation of x+y can lead to any limit.
 
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##\lim\limits_{x^2+y^2\to\infty} x y e^{-(x+y)^2}##

##\lim\limits_{r^2\to\infty} r^2 \cos(\theta) \sin(\theta) e^{-r^2 (1+\sin(2\theta))}##

##\lim\limits_{r^2\to\infty} \frac {r^2 \cos(\theta) \sin(\theta)} {e^{r^2 (1+\sin(2\theta))}}##

The limit is undefined when the ##e^{r^2 (1+\sin (2\theta))}=e^{\infty \cdot 0}## or when ##1+\sin(2\theta)=0## and that happens for ##\theta = \frac{3\pi}{4}, \frac{7\pi}{4}##

##\arctan (\frac{y}{x})=\frac{3\pi}{4}## And that means ##\frac{y}{x}=-1## or along the line ##y=-x## which restricts the domain.
 

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