Problem: I'm trying to verify a claim in this paper where the expression
$$P^\infty_1=1-\frac{1}{2\lambda}(\lambda+\mu+\beta-\sqrt{(\lambda+\mu+\beta)^2-4\lambda\mu})$$
is approximated under three different regimes:
Case 1, $\mu>\lambda$, $\beta(\mu+\lambda)\ll(\mu-\lambda)^2$:
$P^\infty_1\approx \frac{\beta}{\mu-\lambda}$
Case 2, $(\mu-\lambda)^2\ll\beta(\mu+\lambda)$:
$P^\infty_1\approx \sqrt{\frac{\beta}{\mu}}$
Case 3, $\mu<\lambda$, $\beta(\mu+\lambda)\ll(\mu-\lambda)^2$:
$P^\infty_1\approx 1-\frac{\mu}{\lambda}+\frac{\mu\beta}{\lambda(\lambda-\mu)}$
Question: How are these approximations justified?
I've tried to rewriting the square root which gives $$\sqrt{(\lambda+\mu+\beta)^2-4\lambda\mu}=\sqrt{(\lambda-\mu)^2+2\beta(\lambda+\mu)+\beta^2}=\begin{cases}(\mu-\lambda)\sqrt{1+2\frac{\beta(\lambda+\mu)}{(\lambda-\mu)^2}+\frac{\beta^2}{(\lambda-\mu)^2}}\approx(\mu-\lambda)\sqrt{1+\frac{\beta^2}{(\lambda-\mu)^2}}& \text{Case 1}\\\sqrt{\beta(\lambda+\mu)}\cdot\sqrt{\frac{(\lambda-\mu)^2}{\beta(\lambda+\mu)}+2+\frac{\beta}{\mu+\lambda}}\approx \sqrt{\beta(\lambda+\mu)}\cdot\sqrt{2+\frac{\beta}{\mu+\lambda}}& \text{Case 2}\\(\lambda-\mu)\sqrt{1+2\frac{\beta(\lambda+\mu)}{(\lambda-\mu)^2}+\frac{\beta^2}{(\lambda-\mu)^2}}\approx(\lambda-\mu)\sqrt{1+\frac{\beta^2}{(\lambda-\mu)^2}}& \text{Case 3}\end{cases}$$
However, I don't see how to proceed from there. It might be worth mentioning that $\lambda,\mu,\beta$ were defined as $$\lambda=r(1-u_1)\quad \mu=1\quad\beta=ru_1$$ where $r$ can be slightly greater, or smaller, or exactly equal to 1 (e.g. $r\in[0.9,1.1]$) and $u_1$ is positive but very small (e.g. $u_1=10^{-7}$).
Any help or comment is greatly appreciated!