Limits

A limit is the output of a function as it approaches a certain value.

Formal definition of a limit (epsilon-delta)

Let f(x) be a function defined on D (a subset of the real numbers).

$$ \lim_{x\to b} f(x) = L $$ given that for every \epsilon > 0 there is a \delta such that for all \epsilon \in D if 0 < \left| x - c \right| < \delta then \left|f(x) - L\right| < \epsilon

The sandwich principle

Let

$$ f(x) < g(x) < h(x) $$ on the interval [a, b]

if

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

and

\lim_{x\to a} h(x) = L

then

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

is also true.