Showing that a limit exists or showing that it does not exist

[htdc]

Evaluate the limit if it exists or show it does not exist.

limit as (x,y) approaches (0,0) (2x^2y)/(x^2+2y^2)Had this problem on a test and got points taken off - I'm trying to figure out what I did wrong. Obviously when we plug in (0,0) we get 0/0 which isn't allowed.

I let x=y and ended up with (2y^3)/(3y^2) which reduces to (2y)/3. Wouldn't the limit still be 0?

Thanks for the help.

 

[micromass]

What you did is approaching (0,0) along the line x=y. If we approach it along this line, then we indeed find a limit of 0.

But this is not enough. If we want the limit to exist and to be 0, then the limit should exist no matter how we approached it! For example, if I approach it along $y=x^2$, then I should also get 0.

Showing that a limit exists involves more than just choosing a relationship between x and y. We cannot do it like that. We can prove it by definition, or perhaps by using the squeeze theorem.