Limit of 1/x as x approaches infinity Why is $\lim_{x\to\infty} \frac{1}{x}$ equal to $0$, when really the limit appears to be an infinitesimal quantity?
I am trying to understand why there is no distinction between 0 and an infinitesimal quantity in the context of limits.

In the early days of calculus the idea of infinitesimals or "vanishingly small quantities" different from $0$ was used somewhat freely although those doing so knew that they had no rigorous justification for doing so.
At some point mathematicians such as Cauchy and Bolzano tried to treat the notion of limit with more rigor by taming the concept of "infinity" in their arguments.
But it was Weierstrass who finally put this into the form of a definition, the $\epsilon, \delta$ definition which we know today.
Thus mathematicians could speak plainly and say that
$$\lim_{x\to\infty}\frac{1}{x}=0$$
rather than trying to explain what they meant by "the smallest possible number greater than $0$" or other circumlocutions.
In modern times others tried to logically incorporate a notion of "infinitesimals" into calculus in what is called "non-standard analysis."
