Comparing functions via the limit of their ratio

Theory
1

Good evening,
I have a question about section 7.
How can I prove that: n^700 = O(2^n) ??
It seems very obvious, but can I prove it only with the definition of O ??
Thanks in advance,
Almog

2

Your question is a special case of the following (assuming you know that n^{100}/2^n \to 0):
Consider two functions f,g: R^+ \rightarrow R^+.

  1. Suppose that \lim_{n\to\infty} \frac{f(n)}{g(n)} = 0. Prove that f=O(g).
  2. Suppose that \limsup_{n\to\infty} \frac{f(n)}{g(n)} \leq c (where c is a constant). Prove that f=O(g).
3

Yes, I know that the limit is 0.
I thought I can prove n^700 = O(2^n) without relying on this known limit, but only with the definitin of O.
Never mind, thank you.