Mutlivariable Epsilon Delta Proofs

1. Aug 16, 2009

patata

Like many people on this forum, i am seemingly having a lot of trouble grasping the concepts of Epsilon Delta proofs and the logic behind them. I have read the definition and i realise for e>0 there is a d>0 such that...

0<sqrt((x-1)^2 - (y-b)^2) < d then f(x,y) - L <e (excuse my use of proper symbols on this forum...i dont know how!)

The text book has an example lim (x,y) -> (0,0) x^2y/x^2 + y^2 but i am completely oblivious to how they arrive to the conclusion that the limit equals 0!

I can follow it until 0<sqrt(x^2 + y^2)<d then | f(x,y) - 0 | < e but after that my eyes glaze over and im lost no matter how many times i read it. If anybody could walk me through the process of finding these limits, i would greatly appriciate it.

As a side note, i tried reading other threads for help and attempted to convert the questions i tried into polar coordinates but i still couldnt seem to get the right answers doing that =( thanks!

2. Aug 16, 2009

arildno

Now, as a general advice for multi-variable limit study, change into polar coordinates!

Why?

Because the "delta" refers to the DISTANCE between the arbitrary point (say, (x,y)), and the fixed point of study, say (0,0).

But this means that our requirement for the existence of the limit can be reformulated as follows:
"The limit L of f at (0,0) exists if there for every e exists a ball B (with radius d) around (0,0), so that all function values for points in B differs from L with less than e"

In your case, transform x and y by means of
$$x=r\cos\theta,y=r\sin\theta$$
where r is the radial variable.

Therefore, we have:
$$\frac{x^{2}y}{x^{2}+y^{2}}=\frac{r^{2}\cos^{2}{\theta} r\sin\theta}{r^{2}}=r\cos^{2}\theta\sin\theta$$

Now, IF 0 is to be the limit of this expression at (0,0) (in Cartesian coordinates), it means that this must be the limit of the expression as r tends to 0!

We easily see that this is the case.

3. Aug 16, 2009

patata

Thank you for the quick and speedy reply! I understand the case for x^2y but i made a typo and the example for the book is x^3y...

If i convert this to polar coordinates i get

r^3Cos^3(theta) x rSin(theta) / r^2 (Cos^2(theta)Sin^2(theta)

However, i am getting stuck here, are there any ways to simplify Sin(theta)/Sin^2(Theta) or is it just Sin^-1(theta)

P.s. any links on how to format the maths in my posts correctly would be greatly appriciated ^^ they look very sloppy in comparison to the above! thanks for all the help.