# 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.