Definition – Limit of a function at x = c.

Definition

Given a function $f(x)$ , we write

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

If, for every $\epsilon > 0, \exists{\delta > 0}$ with $\epsilon, \delta \in \mathbb{R}$ , $\epsilon$ and $\delta$ fulfill the following condition,

$|x - c| < \delta \Rightarrow |f(x) - L| < \epsilon$

Notes

Firstly, what does this limit tell us? The phrase “if, for every $\epsilon$ there exists an $\delta$ ” means exactly that. Note that epsilon relates to the difference between $L$ and $f(x)$ , then it follows that for every distance from the function’s limit, there is a corresponding minimal distance from $c$ that $x$ can be. This means the behaviour of the function needs to be “if $x$ is within delta of $c$ , then the function is within epsilon of its limit”.