In our study of limits of a function `Lim_(x→a) u(X)`, we defined `x` to exist in an open interval containing a. This means that we assumed that `u(X)` is defined on both sides of a. However, in calculus we also study and evaluate limits w.r.t. one side only(i.e., either for values of x greater than a or those less than a). Such limits are known as One-sided limits.
As evident from the name, one-sided limit can be of two types:
`Lim_(x→a-) u(X)` is defined to be the limit of `u(X)`as x approaches a from the left hand side, i.e., this limit is defined for values of x sufficiently close to a but less than a.
`Lim_(x→a+) u(X)` is defined to be the limit of `u(X)`as x approaches a from the right hand side, i.e., this limit is defined for values of x sufficiently
close to a but greater than a.
As regards the evaluation of one-sided limits, you do not need to be confused about them. All the theorems and solution techniques of limits discussed above
are equally applicable for the evaluation of one-sided limits.
We have gone through the concept of one-sided limits because they are pre-requisite for developing an understanding of a continuous / discontinuous function. Continuity/discontinuity of a function is a topic that you will find frequently in your Mathematics courses, and having a good understanding on the topic will be really worthwhile. We discuss it next.
There are three conditions that need to be met by a function f in order to be continuous at a number a. These are:
If any one or more of these three conditions is not true for `f ` at "a", then the function `f` is called a discontinuous function at "a".