α ≠ 1 is a positive real. Find limx→∞ ( (αx - 1)/(αx - x) )1/x.
Solution
limx→∞ (1/x)1/x = limy→0 yy = 1. If α > 1, then lim 1/(α - 1)1/x = 1, and lim (αx - 1)1/x = lim (αx)1/x = α, so the whole expression tends to α. If α < 1, then (αx - 1)/(α - 1) tends to 1/(1 - α), so the whole expression tends to 1.
© John Scholes
jscholes@kalva.demon.co.uk
4 Dec 1999