limit help

**Kuma** · October 9th 2011, 11:03 AM

hi

how can I evaluate this limit, using the epsilon delta way if possible. I read my textbook about the method but didn't really understand.

lim
(x,y) -> (0,0) of [x-xy+3]/[x^2y+5xy-y^3]

plugging in the point you can see its discontinuous.

so if I move along the x axis in the end I will get x-3/0 and along the y axis I will get -3/y^3, in the end both of those will be -3/0. If I move along the line y = x I also get the same answer. So I can say the limit does nit exist? Now how can I show this using the epsilon delta method for limits?

**FernandoRevilla** · October 9th 2011, 11:46 AM

Firstly: taking into account that $\lim_{(x,y)\to (0,0)}(x-xy-3)=-3$ and $\lim_{(x,y)\to (0,0)}(x^2+5xy-y^3)=0$ , by a well-known theorem $\lim_{(x,y)\to (0,0)}\frac{x-xy-3}{x^2+5xy-y^3}=\infty$ . Then, you can use an "indecent trick": transport the arguments $\epsilon-\delta$ of that theorem to your problem.

**Kuma** · October 9th 2011, 12:16 PM

hi. Thanks for the reply but I'm not sure if i still understand fully. This theorem, is it the limit of a rational function is the limit of the numerator over the denominator? I am also unsure about this trick I can use to apply the epsilon delta method to my problem.

thanks

**FernandoRevilla** · October 9th 2011, 09:10 PM

Originally Posted by Kuma

This theorem, is it the limit of a rational function is the limit of the numerator over the denominator?

In general,

$\lim_{(x,y)\to (x_0,y_0)}f(x,y)=k\neq 0,\;\wedge\; \lim_{(x,y)\to (x_0,y_0)}g(x,y)= 0\Rightarrow$

$\lim_{(x,y)\to (x_0,y_0)}\frac{f(x,y)}{g(x,y)}=\infty$

I mean $\infty$ not $+\infty$ .

I am also unsure about this trick I can use to apply the epsilon delta method to my problem.

Could you transcribe from your book the exact formulation of the problem? That is, does the problem say to find out the limit using the definition?

**Kuma** · October 10th 2011, 12:54 PM

Originally Posted by FernandoRevilla

Could you transcribe from your book the exact formulation of the problem? That is, does the problem say to find out the limit using the definition?

It does not, but I thought it would be the way to start since my book only explains evaluating limits through that method.

However your explanation helped! So I can conclude that the limit goes to infinity.

**HallsofIvy** · October 11th 2011, 07:47 AM

Originally Posted by Kuma

It does not, but I thought it would be the way to start since my book only explains evaluating limits through that method.

That makes no sense. The "epsilon-delta" is a way of proving the limit after you have found it, not a way of evaluating limits.

However your explanation helped! So I can conclude that the limit goes to infinity.

More correctly, the limit is infinity (or simply does not exist).

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