Since $\infty>0$, so $1/\infty >0$, thus I think $1/\infty$ should be infinitesimal, but the calculus book says $$\lim_{x \to \infty} \frac{1}{x}= 0$$
So is $1/\infty$ zero or infinitesimal ?
P.S. I mean are $1/\infty$ and $\lim_{x \to \infty} 1/x$ the same thing here?
On
Your opening comment is correct if one replaces $\infty$ by an infinite hyperreal $H$, say. Indeed if $H>0$ then one also has $\frac{1}{H}>0$ for the infinitesimal $\frac{1}{H}$. This is true by the transfer principle applied to the usual relation $0<x<y\implies 0<\frac{1}{y}<\frac{1}{x}$.
In an infinitesimal-enriched number system, the operator of taking limit as $x\to\infty$ is decomposed into two stages:
(1) evaluating at an infinite value of $x$, say $x=H$; and
(2) taking the standard part, or shadow.
In the example you considered, $\frac{1}{H}$ is infinitesimal, and its shadow is zero. So the answer to your question is, it depends. Before passing to the shadow, it is infinitesimal. After passing to the shadow, it is zero.
The issue you have here that, talking about real numbers, there is no such thing as $\frac{1}{\infty}$. All you can talk about are limits in the sense of $\lim_{x\to\infty}{\frac{1}{x}}$.
Now if you have a converging sequence $(a_n)$, and for every $n$ we have $a_n>c$ for some constant $a$, this does not mean that $\lim{a_n}:=a>c$, all you get is $a\geq c$.
The thing here is that, if you talk about an infinitesimal number in the sense of "an arbitrarily small positive number", you need to be aware that there is no such number.
For every real number $a>0$ there is an even smaller number $\frac{a}{2}>0$, so the concept of one smallest positive number does not make sense. And, keep in mind, that a limit, if it exists, is a number. Not a set of numbers or some obscure thing with questionable properties, it is a number. So in this case you might argue that, in some way, yes, your limit is what comes closest to an infinitesimal positive number, just that that is $0$.
Another way to look at this is one way of looking at $\mathbb{R}$ - now I don't know if you've ever heard of that, but a formal construction of $\mathbb{R}$ can be done by looking at Cauchy-sequences of rational numbers, and saying two such sequences are equal if their difference converges to $0$. Or, in different words, gets arbitrarily small. So, by the construction of $\mathbb{R}$, any infinitesimally small number (talking about absolutes here) is automatically the same as $0$. (Formally speaking, this is a bit lacking, because, well, in our definition we have sequences of rationals, not reals. But since you can put your sequence of reals between two sequences of rationals, that can be remedied.)
So, long story short: There is a way to look at $\lim_{x\to\infty}{\frac{1}{x}}$ as the smallest possible "positive" real, just that in this case you need to be aware that this number can only be $0$ and as such is not positive anymore. Either that, or you take a good look at what the limit actually means, put $0$ into it and directly prove that $0$ is your limit.