The problem statement, all variables and given/known data Prove lim na^n = 0 when 0 < a < 1. The attempt at a solution Without danger, we change from the discrete n to the continuous x so that now we have to prove that lim xa^x = 0. Let e > 0. We have to find an N such that xa^x < e for all x > N. Now if xa^x < e is the same as 1 < eb^x/x, where b = 1/a. Using L'Hospital's rule, we have that lim eb^x/x = lim e(ln b)b^x = oo, so there is an N such that 1 < eb^x/x. QED Is this the simplest way to prove this limit. For some odd reason, I feel that there is a simpler solution. Any tips?