I am reading a proof of the solution to an optimal stopping problem. In this problem, the probability P(M,N) of choosing the best partner when you look at M−1 out of N potential partners before starting to choose one depends on M and N. I can follow the proof to the point where P(M,N) = (M-1)/N * ln((N-1)/(M-2)). The proof goes on to state that for big N this can be simplified to M/N * ln(N/M), but I can't prove why that is so.
I am guessing that the author is taking the limit as N approaches infinity. (M-1)/N goes to zero and ln((N-1)/(M-2)) goes to infinity. Can l'Hopital's rule be applied here? If so, can I treat M as a constant when taking the derivative of, say (M-1)/N and 1/(ln((N-1)/(M-2)))?