Intro to Limits

Recall that a limit is, generally speaking, a value to which a mathematical expression approaches at a certain point. Formally, we can say that the limit of $f(x)$ as x approaches A is L:

If for any given ε > 0 there exists σ > 0 such that |f(x)-L| < ε when 0 < |x-a|<σ

Still with me? I know, that's not an easy definition. Here's what it really means: If the limit exists, as "x" approaches some arbitrary point called "a", f(x) will approach L. The difference between f(x) and L is ε, and we can make it as tiny as we want. No matter how close we want f(x) to be to that limit L, there will exist a σ (the difference between x and a) where f(x) is within ε of L. To give you a crude, casual version, a limit exists if we can find a value L that a function gets really, really close to at a certain point. We can define that "really, really close" part to be as small as possible and still find a value of x to make it work.