Expert: Ahmed Salami - 7/1/2009

Question
Hello,

For some reason or another I can't seem to get the idea of epsilon-delta definitions of limits into my head, specifically multi-variable epsilon-delta proofs.

As an example, one of the problems in my book is...

"Use the e-d definition of limit to show that Lim(x,y)->(0,0) y/(x^2 + 1) = 0."

I understand that I need to relate |f(x)-L| < e to
0 < sqrt(x^2 + y^2) < d, but how to progress further is unclear to me; however, I hope you can help.

Answer
Hi Robert,
Remember that lim (A/B) = lim A/lim B where lim B is not equal to zero.
lim(x,y)->(0,0) [y/(x^2 + 1)] = lim(x,y)->(0,0) y / lim(x,y)->(0,0)(x^2 + 1)  
                             = lim (y->0) y / lim(x->0)(x^2 + 1)  
which can be done separately to achieve your results.

Regards