The idea of limits

To get started, we’ll learn about the concept of a limit or limiting process. We understand limits by using an intuitive approach through the recent example of the first image taken of a black hole.

We can think of the limit of a function at a number (a) as being the one real number (L) that the functional values approach as the (x)-values approach (a), provided such a real number (L) exists.

Let (f(x)) be a function defined at all values in an open interval containing (alpha), with the possible exception of (alpha) itself, and let (L) be a real number. If all values of the function (f(x)) approach the real number (L) as the values of (x(neq alpha)) approach the number (alpha), then we say that the limit of (f(x)) as (x) approaches (alpha) is (L). (More succinct, as (x) gets closer to (alpha), (f(x)) gets closer and stays close to (L).) Symbolically, we express this idea as: (displaystyle lim_{x to alpha}f(x)=L)

A table of values of graphs may be used to estimate a limit.