How can I proof a difficult big o notation?

Question

I can easily understand how to proof a simple big o notation like n⁵ + 3n³ ∈ O(n⁵).

But how can I proof something more complex like 3ⁿ or 2ⁿ ∉ O(n^k)?

Answer 1

Use a proof by contradiction.

Let's prove that 2ⁿ ∉ O(n²). We assume the opposite, and deduce a contradiction from a consequence.

So: assumption: there exists M and n₀ such that 2ⁿ < M n² for all n >= n₀.

Let x be an number such that x > 5, and x > n₀ and 2^x > 4 M. Do you agree that such a number must exist?

Finish off the proof by deducing a contradiction based on the inequality that 2^2x < 4 M x² by assumption.

Now do the analogous proof for k = 3. Then do it for k = 4. Then generalize your result for all k.