Epsilon delta definition of limit

[Ali Asadullah]

Can someone please explain Epsilon delta definition of limit in detail?

 

[HallsofIvy]

I'll give it a shot. You don't say whether you are talking about the limit of a function or the limit of a sequence so I will assume the limit of a function: \lim_{x\to a} f(x)= L if and only if, given any \epsilon> 0 there exist \delta> 0 such that if |x- a|< \delta, then |f(x)- L|< \epsilon.

|a- b| essentially measures the distance between a and b. Saying that |f(x)- L|< \epsilon just says that f(x) is closer to L than distance [math]\epsilon[/math]. And since [math]\epsilon[/math] can be any positive number, that means that we can make f(x) as close to L as we wish, just by making x "close enough" to a.