I need to find the limit of these two fractions as $P$ goes to infinity.
$$v_0 = \frac{-\frac{\mu_k}{P} + \frac{\mu_k^2 c_k^2}{P^2} + s_A^2}{\sqrt{s_A^2 + \frac{\mu_k^2 c_k^2}{P^2}}}$$
$$v_1 = \frac{1-\frac{\mu_k}{P} + \frac{\mu_k^2 c_k^2}{P^2} + s_A^2}{\sqrt{s_A^2 + \frac{\mu_k^2 c_k^2}{P^2}}}$$
By plugging numbers in and graphing it seems that $v_1$ goes to infinity with $P$, but that $v_0$ converges to something finite.
I'd also appreciate it if someone could explain how l'Hopital's rule may or may not be useful in this problem.
Assuming $\mu, c, s$ are constants, all small fractions go to $0$, which makes $$ v_0 \to \frac{s_A^2}{\sqrt{s_A^2}} = |s_A|\\ v_1 \to \frac{1 - s_A^2}{\sqrt{s_A^2}} $$
L'Hospital's rule is not useful here since both numerator and denominator are not of the form $0/0$ nor of the form $\pm \infty/\infty$.