Not all exponential functions increase faster than polynomial. In this case, f9 is the only function out of nine that tends to zero rather than infinity.

Second, not every function that starts with log grows like a logarithm. We have . This is the upper bound. For the lower bound we can use Stigling's approximation , from where for some , and this is from some point. So, and , i.e., . In particular, grows faster than .

Since , we have and . So, grows faster than .