I am currently doing a physics project and have two expressions of two versions of a length $L$ of the form
$$L_h=2\pi N\sqrt{\left\langle R \right\rangle^2 +\left(\dfrac{h}{2\pi N}\right)^2}$$ and $$L_v=\pi N \sqrt{\left\langle R \right\rangle^2+\left(\dfrac{h}{\pi N}\right)^2} +\frac{\pi^2 N^2\left\langle R \right\rangle^2}{h} \ln \left(\sqrt{1+\left(\dfrac{h}{\pi N\left\langle R \right\rangle}\right)^2}+ \dfrac{h}{\pi N\left\langle R \right\rangle} \right)$$
I am trying to find the limit of these expressions as $\dfrac{h}{N}\to0^+$. The issue is that I cannot isolate all $u=h/N$ out of the expressions (there's always one $N$ as a coefficient on each term.)
When I use MATLAB to calculate the limit, I get $$\lim_{h\to0^+}L_f=2\pi N\langle R\rangle$$ and $$\lim_{h\to0^+}L_v=\frac32\pi N\langle R\rangle$$
However, I'm not sure about
whether $\lim_{h\to0^+}L = \lim_{h/N\to0^+}L$, and
how to prove these results mathematically.
Thank you very much.
$h$ seems to be some kind of a length-scale of your system, thus you may try to rescale $L_\nu$ by it, i.e. devide all your equations by $h$ and you have $u$-s all around. As for the rest: expand the root and the log consecutively in a power-series, then you may easily carry out the limit. For instance: $$ \sqrt{\langle R\rangle^2 + \left(\frac{h}{\pi N}\right)^2} \to \langle R\rangle +\frac{1}{2\langle R\rangle}\left(\frac{h}{\pi N}\right)^2 $$