Definition of limit of a function at infinity

Question

I looked at some rigorous definitions of limit of a function at infinity (in real analysis). The one I found most often is this:

Why Open Interval In Formal Definition Of Limit At Infinity

I will assume the +∞ case below.

I haven't been able to figure out why is it required that the function be defined in an interval (a,∞).

Won't it be sufficient just to say that for any c∈ℝ, there exists m∈ℝ, m>c such that the function is defined at m (the domain of the function has no upper bound)? Such definition will be less restrictive and can directly define limit of a sequence. Is there something wrong with that definition?

Answer 1

You can do that too, but the concept it produces is usually not the one we pragmatically want.

Intuitively we'd like, by analogy with the finite case, to say that $L$ is the limit iff: For every $\varepsilon>0$ all we have to do to guarantee $|f(x)-L|<\varepsilon$ is to choose $x$ sufficiently close to $\infty$ (that is, large enough).

With your proposal it wouldn't be sufficient to make $x$ large enough; we would additionally need to avoid $x$s where $f$ is not defined.

Dealing with this is usually enough extra work, for so little practical gain, that it is not the favored way of doing things and is not what the short ordinary notation gets to be used for. But again, your limit concept is available when you need it; you just need to specify something like $\lim\limits _{x \to\infty, x\in A}$ where $A$ is the upwards unbounded set you're interested in.