# what's the difference between the limit as x approaches zero and an infinitesimally small number [closed]

When x approaches zero doesn't it become infinitesimally small or in other words become an infinitesimal? I was told that they are distinct concepts,for example wikipedia states that there two types of calculus standard and non standard, and in non-standard calculus infinitesimals are used instead of limits. wikipedia also states that the idea of limits resolved many debates on the logical validity of infinitesimals.

• 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
• 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. – user856 Aug 22 '19 at 13:30
• Possible duplicate of Are infinitesimals equal to zero? – user856 Aug 22 '19 at 13:30
• Why did you ask exactly the same question as vipin rawat but 9 hours later? math.stackexchange.com/questions/3330639/just-perception – Floris Claassens Aug 22 '19 at 13:45

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 $$0<|\omega|. i.e. $$\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.