Calculate the limit? 
The answer sheet says:

I understand all of this apaprt from the last bit where it says 
I don't understand how/why they have put logs and exponentials in there. 

The basic result needed
is that
$\lim_{x \to \infty} \dfrac{x^na^x}{b^x}
= 0
$
for any
$n > 0$
and
$0 < a < b$.
Essentially,
an exponential function
of the form
$c^x$ where
$c > 1$
eventually grows faster
(for large enough $x$)
than any fixed power of $x$,
$x^n$,
no matter how large $n$ is
and how close to $1$
$c$ is.
