In today’s blog post, I want to give an answer to the following question: what is the precise mathematical definition of a limit?

For instance, what do I actually mean when I say that the limit of f(x) as x approaches a is L? Well, the first thought that may have come to your mind is that f(x) “approaches” L as x “approaches” a. But, what is the meaning of “approach”?. for example, if I let f(x) = x² and I say that x gets closer and closer to 0, then one may also argue that f(x) is “approaching” -1, as while f(x) gets closer and closer to 0, it also gets closer and closer to -1. However, we know that this is not the case.

So, how do we answer this question? Well, one way of thinking about this is that f(x) should get “arbitrarily close” to L, i.e. that no matter how small a number n I choose, be it 1/100, 1/1000 or even 1/10¹⁰⁰, there exists a sufficiently small interval around a such that if x gets close enough to a so as to belong in that interval, then the difference between f(x) and L is smaller than n. With this idea, we have stumbled upon the true meaning of a limit in mathematics, and below, I give the formal definition of a limit as is used in real analysis which is based on the idea developed in the previous sentence.

Let f(x) be a function defined on an open interval around x = a (note that f(a) need not be defined). We say that the limit as x approaches a is L if for every ε > 0, there exists δ > 0 such that for all values of x, 0 < |x – a| < δ implies that |f(x) – L| < ε.

I hope you enjoyed pondering the question which I addressed in this blog post!!