limit

A limit is the value that a sequence or function can get within an arbitrary distance of.

Limits exist in any metric space.

The limit of a sequence is the value which the sequence "tends to", i.e. as the index increases, the sequence elements get closer to the value of the limit.

Formally,
the limit of a real sequence $(x_{n})$ is

lim_{n \to \infty} x_{n} = x

where $x$ a real number such that for any real $ε > 0$ , there exists a natural number $N$ such that for every natural $n > N$ , then $| x_{n} - x | < ε$ .

The limit of a function is the value which the function "gets arbitrarily close to", i.e. as the input gets closer to the point of interest, the output gets closer to the limit.

Formally,
the limit of a real function $f$ at $c$ is

lim_{x \to c} f (x) = L

where $L$ is a real number such that for any real $ε > 0$ , there exists a $δ > 0$ such that for any $x$ with $0 < | x - c | < δ$ , then $| f (x) - L | < ε$ .