If $\lim f(x)$ can theoretically be anywhere on the interval $0$ up to and including $\infty$, is the interval written $(0,\infty]$? Let's take $f(x)=a^x+b$ where $a\in\mathbb R$ and $b\in\mathbb R^+$. clearly $L=\lim_{x\to\infty}f(x)>0$, but is the interval written $L\in(0,\infty)$ because limits only approach infinity or is $L\in(0,\infty]$ because infinity is one of the values $L$ can take? I can't find an example of this one way or another.

If you want to be able to write $L = \lim_{x\to\infty} f(x) = \infty$, then it would be false to state $L \in (0,\infty)$.
You would need to be able to write $L \in (0,\infty]$, but we also need to keep in mind that $L = \infty$ isn't a real number, so this only makes sense in the context of the extended real number line.
