Question

what is limit of sequence...?

Answer 1

Let \[\varepsilon > 0\] be arbitrary. Then \[\exists N \in \mathbb{N} : |a_n - l| < \varepsilon \hspace{0.3cm} \forall n > N\]
There is a dependence of \[\varepsilon\] on \[N\] and it's written as \[\varepsilon = \varepsilon (N).\]
In the above, \[a_n\] are just terms of your sequence for \[n>N.\]

Answer 2

\[l\] being the limit of your sequence. If you are given that \[l\] exists for your sequence, then the above is true by definition.

Answer 3

actually more properly, \[N=N(\varepsilon )\] is the dependence as epsilon is arbitrary.