Does $\infty$ mean $+\infty$ in "English mathematics"?

My question background is the set $\mathbb R$.

I often see the symbol $\infty$. Does it always mean $+\infty$ or can it have the meaning of $\pm \infty$? In particular what means that a real sequence $(a_n)$ has $\infty$ for limit? That it has $+\infty$ for limit? Or that the sequence $(\vert a_n \vert)$ has $+\infty$ for limit?

I'm French and we usually don't use $\infty$ without a sign in the context of real numbers.

Solution 1:

In the context of real numbers it is a relatively safe assumption that an author writing that a sequence $(x_n)$ tends to $\infty$ means that for each $M>0$ there is an $n_0$ such that $x_n \ge M$ for $n \ge n_0$, and not $|x_n| \ge M$ for $n \ge n_0$. If the latter is meant it would usually be indicated.

In the context of complex numbers it will of course be different.

Solution 2:

If you write $\lim\limits_{x\to\infty} f(x)$ and $x$ is understood to be real, then it means $\lim\limits_{x\to+\infty} f(x)$ unless there is some context saying it means something else. If you write $\lim\limits_{z\to\infty} f(z)$ where $f$ is a complex-valued function of a complex, rather than real, variable $z$, then it means neither $+\infty$ nor $\pm\infty$. Rather it essentially means the limit as the absolute value $|z|$ approaches $+\infty$. In other words $\text{“} \lim \limits_{z\to\infty} f(z) = c\text{”}$, where $c$ is some complex number means $f(z)$ can be made as close as desired to $c$ by making $|z|$ big enough. Or more precisely: $$\forall \varepsilon>0\ \exists R>0\ \forall z\in\mathbb C\ \big[ \text{if } |z|>r \text{ then } |f(z)-c|<\varepsilon \big].$$

Sometimes one writes $\lim\limits_{x\uparrow\pi/2} \tan x = \infty$ or $\lim\limits_{x\uparrow\pi/2} \tan x = +\infty$ and $\lim\limits_{x\downarrow\pi/2} \tan x = -\infty$, but I think it makes sense to write $\lim\limits_{x\to\pi/2} \tan x = \infty$ where $\text{“}\infty\text{''}$ means the infinity that is at both ends of the line and is approached by going in either direction. That makes the tangent function continuous everywhere on $\mathbb R$. But I would accompany this with an explanation of the intended meaning.