Is this a limit, or am I using the wrong word? I am looking at an equation that says:
$$z_\frac{1}{1000}(a, \lambda) = \frac{\ln(1000)a\lambda}{2\pi\sqrt{(\chi'_{mn}\lambda)^2 - a^2}}$$
What I want to say is something like, "as a becomes very small, the function becomes approaches being tangential $\frac{\ln(1000)a}{2\pi\chi'}$, where 'very small' means a small fraction of lambda". I'm pretty sure that's true, but I don't know if that's the same as saying $$\lim_{a \to 0} z_{\frac{1}{1000}}(a, \lambda)= \frac{\ln(1000)a}{2\pi\chi'}$$
It's been a long time since I took limits, and I'm not sure it's legal to put the variable in the limit on the left-hand side, too. Does this make any sense?
(There are a bunch of physics conditions involved in the specific problem enforcing that 0 < a < $\chi'_{mn}\lambda$)
 A: You are correct, this is not a limit.
This is because the limit as $a\to 0$ can never have $a$ in it - the limit as $a\to0$ can never depend on $a.$
One way to think of $\lim_{a\to0} f(a)$ is as the answer to the question, “What value should we give $f(0)$ to make $f$ continuous at $0?$” What would $a$ even mean in formula for the limit? $a$ isn’t even a variable in that question.
If $$\lim_{a\to 0} g(a)=\lim_{b\to 0} g(b),$$ so if the limit is $2a,$ does $2a=2b?$ No.
You can write this instead by writing,

Near $a=0,$ $$z(a,\lambda)\sim \frac{\ln(1000)a}{2\pi\chi’}.$$

Or, equivalently, you can say:
$$\lim_{a\to0}\frac{z(a,\lambda)}{a}=\frac{\ln(1000)}{2\pi\chi’}$$
In general, we say $f(a)\sim g(a)$ near $a=a_0$ if $$\lim_{a\to a_0}\frac{f(a)}{g(a)}=1.$$
Often, when the context makes it clear, we skip the phrase “near $a=a_0.$”
I usually read $\sim$ as “is asymptotic to.” Wikipedia says the name is “on the order of” and the description is “$f$ is asymptotically equal to $g.$”
See the “Generalizations and related usages” and “ Family of Bachmann–Landau notations” sections here.
