>>7031556The idea is that for any possible ε>0, usually small but can be of any size really, you can find some δ such that |f(x,y)-l|<ε (l being the value of the limit) whenever the condition 0<√((x-a)2+(y-b)2)<δ is satisfied. In this case a,b,l=0.This is the definition of a multivariate limit, and is analogous to the single variable definition of a limit, did you skip single variable calculus or something? The main idea is to manipulate |f(x,y)-l| to contain √(x2+y2) as this is <δ, which may be chosen by us to then force the entire thing to be less than the given ε. δ is a variable within our control, whilst ε is not. Put even more informally, you claim the limit as (x,y)(0,0) is 0, but nobody believes you so you're being asked "make f(x,y) within ε of l, by approaching (0,0)" and you reply with "f(x,y) is always within ε of l when (x,y) is within δ of (0,0)" where δ is your answer to the question. It's clear δ will almost always have a dependence on ε, except for constant functions and maybe some other weird exceptions.