Show that if x epsilon R limn rightarrow infinity xnn 0Solu

Show that if x epsilon R, lim_n rightarrow infinity x^n/n! = 0

Solution

Since this is a limit, we don\'t care about any finite value. Choose to look at the case of n = x. That is equal to x^x/x!, a constant. With each increase of n, you multiply that constant by x/n, and we know that n > x now, so that factor is < 1. As n --> infinity, the product of those will tend to 0. Thus, the limit = 0.