What does some function evaluated at $0^+$ mean?

Question

Let's say there's an expression like this:$$a = p(x)|_{x=0^+}$$ Is it the equivalent of $$a=\lim_{x\to 0^+}p(x)$$ or does it/can it mean something else?

Answer 1

I've seen it used in two contexts, so you might have to match this with what your situation is. Generally its used more in a physics sense than a pure mathematical one.

The first interpretation is what you described $$ p(0^+) = \lim_{x\rightarrow 0^+} p(x) $$

which is equivilent to the second definition. This second definition is certainly more loose, but has its usages in electrical engineering, physics, etc.

$p(0^+)$ is the value of $p(t)$ at the instant immediately after $t=0$

I've personally seen this used in the context of electrical signals when a switch is flipped on "the instant after" $t=0$, which I'll admit doesn't make much sense, but is sufficient enough to solve the problem at hand