## Which function is dominant?

Exponential grows faster than any polynomial.
$\lim e^x / x^n = \lim e^x / (n!) = \infty$ when $x \rightarrow \infty$.

Logarithm grows slower than any polynomial.
$\lim \frac{\ln x}{x^{1/n}} = \lim n \frac{1}{x^{1/n}} = 0$ when $x \rightarrow \infty$.

This might surprise you, $\ln x$ seems to dive into $- \infty$ pretty fast when $x \rightarrow 0$, but still not fast enough to be dominant:
$\lim x \ln x = \lim (\ln x)/(1/x) = 0$ (L’Hospital’s rule).