What is the difference between the limit as $x$ approaches zero and an infinitesimally small number?

Question

When $x$ approaches zero, doesn't it become infinitesimally small or in other words, an infinitesimal?

I have been told that limits and infinitesimals are distinct concepts. For example, Wikipedia states that there are two types of calculus : standard and non standard.

It states that in non-standard calculus, infinitesimals are used instead of limits, which are used in standard calculus. Wikipedia also states that the idea of limits resolved many debates on the logical validity of infinitesimals.

I am unable to understand how infinitesimals and limits are different.

Sentenced start with a capital letter and end with a period. You put no effort into your question, yet you expect effort from us to answer it. — Klangen, Aug 22 '19 at 13:16
Your question lacks context. Where does this problem come from? Who told you there was a different and in what situation was this in? — Toby Mak, Aug 22 '19 at 13:19
In addition this question is almost exactly the same as what you are asking. — Toby Mak, Aug 22 '19 at 13:20
The limit of $x$ as $x$ approaches zero is exactly zero, not an infinitesimally small number. There are no infinitesimally small numbers in the real number system. — , Aug 22 '19 at 13:30
im sorry im very new at this, please excuse me forgetting a period her and there. — crazyfoo, Aug 22 '19 at 13:31
Why did you ask exactly the same question as vipin rawat but 9 hours later? https://math.stackexchange.com/questions/3330639/just-perception — Floris Claassens, Aug 22 '19 at 13:45
Complete coincidence. Please don’t lash out at me I just wanted to ask a question. — crazyfoo, Aug 22 '19 at 14:19
I don't understand why this question was closed. It makes sense to me and conveys everything pretty clearly... — Rajdeep Sindhu, Jan 16 '21 at 13:05

score 1 · Answer 1 · answered Aug 22 '19 at 13:42

An infinitesimal is an infinitely small number. One way you could define this is that $\omega$ is an infinitesimal if for all real numbers $r\in\mathbb{R}_{>0}$ we have $0<|\omega|<r$. i.e. $\omega$ is non-zero, but it's absolute value is smaller than any positive non-infinitesimal number.

A sequence $(x_{n})$ in the real numbers converging to $0$ becomes arbitrarily small, not infinitesimally small. So for every real number $\varepsilon\in\mathbb{R}_{>0}$ we can find some $N\in\mathbb{N}$ such that for all $n\geq N$ we have that $|x_{n}|<\varepsilon$. Note though that every sequence element $x_{n}$ is still a non-infinitesimal numbers (except if $x_{n}=0$, but that does not really count in this context).

That being said one way to construct infinitesimal numbers is to look at certain equivalence classes of sequences converging to $0$, so your intuition that a sequence converging to $0$ is somehow linked to an infinitesimal number is not completely wrong.

What is the difference between the limit as $x$ approaches zero and an infinitesimally small number?

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