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 standard real analysis/calculus, there are no infinitesimal quantities. Everything is formulated in terms of real numbers. What $\lim_{x\to \infty} f(x) = c$ means is that for all $\varepsilon > 0$ there exists $x_o\in \mathbb{R}$ such that whenever $x>x_0$, we have that $\vert f(x)-c\vert < \varepsilon$. In words, what this means is that if you pick and small number $\varepsilon$, you can find a number $x_0$ large enough such  that at any number past $x_0$, $f(x)$ will be no greater than a distance $\varepsilon$ away from $c$. For example, if $f(x)=\frac{1}{x}$ and $c=0$, this is the case. Given $\varepsilon > 0$, we can let $x_0 = 1/\varepsilon$; then if $x>x_0$, we have $\vert f(x)-0\vert = 1/x < \varepsilon$. So, $1/x$ really does approach $0$, in that it gets arbitrarily close to $0$.
