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 \(\displaystyle \epsilon\). And since \(\displaystyle \epsilon\) 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.