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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.

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    $\begingroup$ 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. $\endgroup$ – Klangen Aug 22 '19 at 13:16
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    $\begingroup$ Your question lacks context. Where does this problem come from? Who told you there was a different and in what situation was this in? $\endgroup$ – Toby Mak Aug 22 '19 at 13:19
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    $\begingroup$ 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. $\endgroup$ – user856 Aug 22 '19 at 13:30
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    $\begingroup$ Possible duplicate of Are infinitesimals equal to zero? $\endgroup$ – user856 Aug 22 '19 at 13:30
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    $\begingroup$ Why did you ask exactly the same question as vipin rawat but 9 hours later? math.stackexchange.com/questions/3330639/just-perception $\endgroup$ – Floris Claassens Aug 22 '19 at 13:45
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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.

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