How can I prove that eventually one function will overtake another, and find when? Given for example 2 functions,$\ n^{100} $ and$\ 2^n$. I know that$\ 2^n$ grows faster and that therefore there is some$\ n$ where it will eventually overtake $\ n^{100} $ but how can I prove this, and also maybe find that$\ n$?

Let $\lambda > 0$ and consider the limit
$$\lim_{x\to\infty} {x^n\over e^{\lambda x}}.$$
This limit can be reduced using L'hospital's rule to 
$$ \lim_{x\to\infty} {nx^{n-1}\over \lambda e^{\lambda x}}.$$
Repeating, the power whittles away whilst the exponential remains on the bottom.
