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.

Technically the answer is neither.
When we say $x \in (0, \infty)$, that notation is only valid when $x$ is a number.
When we say $\lim_{x\to\infty} = \infty$, that does not mean the limit is the number $\infty$. It means $\forall N> 0: \exists M> 0:$ if $x > M,$ then $f(x) > N$.
However you could extend the definition of the bracket $[a,b]$ notation to allow for such limits that do not take any value.
