Limit involving delta-epsilon proof help

[ZPlayer]

Hi, Everyone,

Problem asks to prove that limit of x^2 * sin^2 (y) / (x^2 + 2* y^2) as (x,y) approach (0,0) is 0 using delta-epsilon method. I think I solved it below. Let me know if I am write or wrong.

Thanks.

 

[ZPlayer]

I am sorry for the previous post. The solution I have arrived at is actually below.

 

[Mark44]

Something easy you can do is pick some numbers to see if it works. Pick a value for epsilon that's fairly small, such as 0.1. Now pick a point (x, y) that is within delta (= epsilon) of (0,0). Is the value of your first expression less than epsilon?

Now pick a smaller number for epsilon, such as 0.01, and repeat the process as above. This doesn't prove that you have selected the right value of delta, but if you find that the process doesn't give the results you want, then you know you haven't found the right delta.

 

[ZPlayer]

I understand your explanation. However, I would appreciate it if you could go over the logic of my solution and answer the question conclusively (i.e. to the best of your knowledge) if my solution is correct or not. Thank you for your time and effort.