Limit of multi-variable

[hazellaw]

i need some help with this question

Find the limit, if it exists, or show that the limit does not exist

lim_(x,y)->(0,0) x²sin²y/(x²+2y²)

i've tried to x=y x=0 or x=y² but i still got 0...

 

[jostpuur]

With these kind of exercises, there are two alternatives:

It could be that the limit does not exist, and then you can prove it by finding two different paths to the origo, such that the limit is different along them.

Or then, it could be that the limit does exist,and then a one good idea is to use polar coordinates, or in some other way obtain some bounds (upper or lower, whatever you need) as a function of $r$ (distance from origo), and then prove that the bounds converge when $r\to 0$.

If you don't know in advance what the correct solution is, then you must try both to see what works.

I believe this expression approaches zero when $(x,y)\to 0$, but it could be I made mistake.

 

[tiny-tim]

Hi hazellaw!

Can you see an easy way of doing it for x²sin²y/(x² + y²) ?

Then adapt that.