Calculus (II): What Is A Limit? How To Really Grasp It?

Did you know that a derivative is actually a limit?

The Limit — Illustration created by the author

Every beginning calculus student faces the following challenging question: What is a limit? The notion of the limit is arguably the second most important one to grasp after that of a function. In my first entry in the calculus series, I covered the ground fundamentals of a function.

This essay is the second entry in the calculus series, where I will be diving into a deep discussion about the notion of a limit. If you did not know or realise, the typical derivative from differential calculus is actually a limit. And so is the typical integral from integral calculus.

So, before even starting out with derivatives and integrals, it makes absolute sense to strengthen one’s fundamental understanding of the limit. Having said this, I mentioned “typical derivative” and “typical integral” for a reason.

You see, calculus exists in two prominent variants: the way schools teach calculus (the conventional way) and the Nonstandard variant. The latter does not use the notion of the limit whereas the former does.

In this essay, we will only be discussing conventional calculus (don’t worry, I’ll cover the Nonstandard variant in a fresh essay). Without any further ado, let us begin.