16th Putnam 1956 Problem A1 α ≠ 1 is a positive real. Find lim x→∞ ( (α x - 1)/(αx - x) ) 1/x .
lim x→∞ (1/x) 1/x = lim y→0 y y = 1. If α > 1, then lim 1/(α - 1) 1/x = 1, and lim (α x - 1) 1/x = lim (α x ) 1/x = α, so the whole expression tends to α. If α < 1, then (α x - 1)/(α - 1) tends to 1/(1 - α), so the whole expression tends to 1. 16th Putnam 1956 © John Scholes jscholes@kalva.demon.co.uk 4 Dec 1999