Calculus (IV): How To Replace Limits With Infinitesimals?

Practically LIKE zero but conceptually NOT zero!

lim x →0, Sin(x)/x — Illustration created by the author

While mainstream calculus uses the concept of limits, an alternative version differs to replace limits with infinitesimals. And it is just as effective. Some mathematicians even report that this alternative version solves certain problems faster as compared to the mainstream version.

In my essay on the notion of limits, I had mentioned that I will be covering the infinitesimal calculus in a separate essay. So, here we are. This also happens to be the fourth entry in my calculus series.

I will begin by giving you a brief historical journey of infinitesimal calculus. Following this, I will proceed to cover the intuitive difference between calculus using infinitesimals and limits.

Finally, I will touch upon the advantages and disadvantages of using this alternative version practically. Without any further ado, let us begin.

What are Infinitesimals?

It is easy for any modern calculus practitioner to think that calculus has always used the notion of limits. But this was certainly not the case. In fact, the inventors of calculus never used limits; they used infinitesimals.