The Limit

If we let 𝑓(x) be a function and x0 be a particular real number, then the following expression may be interpreted in several ways.

In some sense, this is likely the first operation you will see that cannot be expressed in terms of elementary operations. We will review limits in terms of a intuitive geometric interpretation, in terms of a methodology to numerically approximate such expressions, in terms of an algebraic procedure and in terms of a topological expression regarding distances. Each consecutive interpretation will be a little more abstract and difficult to understand but will yield a broader understanding of the underlying concept.

Geometrically

The limit can be imagined as a tiny creature walking along the curve traced out by 𝑓(x) towards x0 to meet its cohort walking from the opposite direction. If the two tiny creatures can meet and shake hands while walking along the curve, then we say the limit exists. We say that the limit is equal to the height at which they meet. If there is a difference in height, however small, our two tiny creatures cannot reach out and shake hands and we say the limit fails to exist.

This naturally brings about the idea of one-sided limits. Even when the limit fails to exists, often, a one-sided limit exists. All this means is that our tiny creature can walk from a nearby x to x0. If our creature is walking from the left, we write (the limit from the left) and if we are examining a limit from the right then we write . If the limit from the left does not equal the limit from the right, then we say the limit does not exist. The only way a one-sided limit may fail is at an isolated point or when a function is not defined over a region.

For instance, does not exist, but