If f = O(g), is e^f = O(e^g)?

Question

If f = O(g), is e^f = O(e^g)?

I'm having difficultly figuring out the above question. An example would be welcome. Also, if you use l'Hôpital's rule, please show how you do differentiation.

Answer 1

This statement is wrong, for example 2n = O(n), but exp(2n) != O(exp(n)). (The latter would mean exp(2n) <= C exp(n) for sufficiently large n, i.e. exp(n) <= C which is not true.)