On reading the notation $x\to \infty$

Question

Why is it better to read $x\to \infty$ as "$x$ grows without bound" rather than "$x$ approaches/comes closer to infinity"?

Answer 1

This comes down to the "$\infty$ is not a number" problem.

Here's one argument against saying $x$ "approaches infinity": if $x\to c$ for some number $c\in \mathbb R$, then we say that $x$ approaches $c$ in that $|x-c|\to 0$. However, when $x\to\infty$, we can't say that $|x-\infty|\to 0$. So, $x$ doesn't really "approach" infinity in the same sense.

Answer 2

You can pronounce it however you want -- the important thing is the definition being referred to. And "$x\to\infty$" alone doesn't even have a (commonly used and accepted) definition; only larger pieces of notation of which it is part get ones, such as $\lim_{x\to\infty} \cdots x\cdots$.

There are some formalisms where one considers "infinity" to be an actual point in a topological space that it makes excellent sense to approach. Some of these are consistent with the usual use of "$x\to\infty$" in elementary real analysis. It is often mostly up to temperament whether you use one of them or prefer to reduce everything to talking about actual numbers only.

Answer 3

Because "$x$ approaches infinity" assumes the existence of an object called infinity that $x$ approaches. While there's no big deal with in deed adding such a symbol (it's just an object with a name, not something mysteriously supernaturally big), one should still take care as $\lim_{x\to\infty}f(x)=a$ is still not defined the same way $\lim_{x\to c}f(x)=a$ is: There is no point in saying "For all $\epsilon>0$ there is a $\delta>0$ such that for all $x$ with $|x-\infty|<\delta$ (?!) we have $|f(x)-a|<\epsilon$"