Studying limits and continuity of multi variabled functions

[tomelwood]

Homework Statement

I have a couple of related questions on this topic which are causing confusion at the moment!
a) Study the limit at the origin of: (xy^2)/(x^2+y^4)

b) Study the continuity at the origin and the existence of the iterated limits at the origin of:
i) f(x,y) = { x^2 sin(1/y), y=/= 0
{ x^2 , y=0
ii) f(x,y) = { (x^2 y^2)/(x^2 y^2 + (y-x)^2), (x,y)=/=(0,0)
{ 0 , (x,y)=(0,0)
(I hope it is clear notation in b) to see that these functions take one value for certain parameters, and another for a different parameter.)

Homework Equations

The Attempt at a Solution

a) My first thought here was converting it to polar coordinates, ((r,T) for typing ease) but this made the expression rather more horrendous: f(r,T) = r^3cosT(sinT)^2/(r^2(cosT)^2 + r^4(sinT)^4 which I don't think is helpful in any way, unless I am missing some handy trig identities.
I then thought about going from the definition and took the modulus of the expression. Now, is it true that |f(x,y)| <= |x|? If so then I am done as that tends to 0 and there is my limit. We are supposed to study iterated limits as well, but these seem to be undefined here (giving 0/0 each time, don't they?) so I haven't done that.

b)i)The iterated limit as x-->0 is 0, but the limit as y->0 is undefined, or so I believe. (although multiplying top and bottom by y gives the limit as yx^2, which ->0 as y->0?) To find out if it is continuous, study the limit. As the iterated doesn't exist, study the double/'normal' limit. I don't really know where to go from here, to be honest. But if that bracket is true, then the iterated limit is zero, but the value of the function at zero is x^2 so it is not continuous there?

ii)I'm afraid I really don't know what to do here. Conversion to polar coordinates looks fruitless and the iterated limits are undefined at the origin.

I hope I have given enough information and attempts! This is my first post at Physics Forums, so I apologise for any omissions.
Thanks in advance.

 

[micromass]

Conversion to polar coords is in this situation almost never a good idea. It just makes things complicated.

For the first limit, consider the sequence $$(\frac{1}{n^2},\frac{1}{n})$$. Does the image of this sequence converge?

Try solving the rest by this method...

 

[tomelwood]

Um, substituting in x=1/n and y=1/(n^2) gives n^5 + n^-1 which does converge as n->0 so this limit exists and is zero.

The second one gives the sint/t identity, so has a limit 1, as n->0 but the limit at the origin is x^2 so this is not continuous. What about iterated limits here? As the question says "study continuity and iterated limits"

The third gives a polynomial on the top and bottom : n^6 +x^4 -1.5x^3 / x^6
So a poly on top and a poly on bottom means both cts, and converge to limit 0 so is cts at 0 (l'hopital here I imagine.)

Just realized I've done (1/n^2 , 1/n) but it appears to work better that way for bi anyway. Also, can I just ask what the motivation is behind using this method, as I would never have thought to do that!

 

[micromass]

This method is only for proving that a limit doesn't exist. If you find a sequence that converges to 0, but the image of that sequence doesn't converge to 0, then the limit doesn't exist.

To show that the limit does exist, requires a bit more creativity...