I have a question about the following statement from an article here. He states on page 490 that:

https://sci-hub.ru/10.1002/mana.201200319 

Let $\alpha = \dfrac{\rho_1\lambda^2}{\kappa_1 + i\kappa_2\lambda}$ where $\rho_1, \kappa_1, \kappa_2 > 0$. then 
$$
\operatorname{Coth}(i\alpha L) \to 1, \ \ \text{ when } \ \ n \to \infty 
$$
where $L > 0$. 

Well, firstly, I think he's changing $\alpha$ to $\alpha_{n}$ and leaving $n$ hidden. And doing some mathematical manipulations, I'm getting to 

Let $x_n = \Re(\alpha_n L)$ and $y_n = \Im(\alpha_n L)$. Using some properties, I can get
$$
\operatorname{Coth}(i\alpha_n L) = \dfrac{\cos(x_n)\sin(x_n)}{\cosh^2 (y_n) - \cos^2(x_n)} - \dfrac{\tanh(y_n)} {\frac{\cos^2(x_n)}{\cosh^2 (y_n)} - 1}
$$

From the equality above can I conclude what I want? Why?