what is the limit to infinity of $a^x/x$

Question

I need help with proving that $$\lim_{x \to \infty} \frac{a^x}{x} = \infty$$

I understand the logic but cannot seem to fit the epsilon-delta definition to it.

thanks

note that you should require $a > 1$ for this to work – Ant May 09 '14 at 18:00 — Ant, May 09 '14 at 18:00

score 1 · Answer 1 · answered May 09 '14 at 18:01

1

Hint If $n$ is integer and $a>1$ you have

$$a^n =(1+(a-1))^n > \frac{n(n-1)}{2} (a-1)^2$$

Use this, and the fact that there exists an integer $x-1 <n \leq x$ to conclude that for $x$ real you have

$$a^x > \frac{(x-1)(x-2)}{2} (a-1)^2$$

answered May 09 '14 at 18:01

N. S.

thanks, but why is (1+(a−1))n>n(n−1)2(a−1)2? – user148005 May 10 '14 at 07:39
@user148005 $$(1+(a-1))^n =1+\binom{n}{1} (a-1)+\binom{n}{2}(a-1)^2+.... > \binom{n}{2}(a-1)^2$$ – N. S. May 10 '14 at 18:31

1 Answers1