Another way to think of infinity is as follows. One way to construct the real numbers from the rationals is via Cauchy sequences, i.e. rational numbers "converging" (in effect) to the real number. For example, what is the meaning of $\pi$? Given a list of digits $3.14159\ldots$, we may think of $\pi$ as a sequence of rational converging to $\pi$, for example $$3, 3.1, 3.14, 3.141, 3.1415, 3.14159, \ldots$$
We can do the same with infinity! Let's an infinite number as a sequence diverging to infinity. One example is $$1,2,3,4,5,6,\ldots$$ We can then define all the usual arithmetic operations as usual, for example $$1,2,3,4,5,6,\ldots + 1,2,3,4,5,6,\ldots = 2,4,6,8,10,12,\ldots$$ There is now a very concrete meaning of "the limit of a function $f$ at $1,2,3,4,5,6$" - that is just the limit of the sequence $f(1),f(2),f(3),\ldots$. If a function converges to some value at $\infty$, then it will converge to the same value under all divergent sequence. Compare this to the definition of limit at a point $x$ - we should get the same limit whatever the sequence converging to $x$ is.
Things become more complicated when we try to understand when $a < b$ for two infinite numbers. If $a,b$ are finite and represented by some sequences $a_i,b_i$ converging to them, then if $a<b$ we know that eventually $a_i < b_i$. However, that is not the case with infinite sequences - consider for example $1,2,3,4,5,6,\ldots$ vs. $0,4,0,8,0,12,\ldots$. This can be fixed using some "decision rule" implemented using an ultrafilter, which I'm not going to explain here.
Having done this, we have a full fledged number system, and for each infinite "number" $\alpha$ we can form a corresponding infinitesimal $1/\alpha$ which is positive but smaller than any "real" number (i.e. smaller than any constant sequence $x,x,\ldots$ where $x$ is real).
If you find this interesting, look up nonstandard analysis.