Q:

Limit of p^n*log(p) as p->0

I am unsure how to compute the limit of (p^n)*(log(p)) as p->0. 
I tried the l'hopital rule, but this does not seem to help (for a general n) 

A:

if $n<0$ then
$p^nln(p)=\frac{ln(p)}{p^{-n}}$ and the limit is $-\infty$.
if $n=0$ then
$p^nln(p)=ln(p)$ and the limit is $-\infty$
if $n>0$
$p^nln(p)=\frac{1}{n}p^nln(p^n)=$
$\frac{1}{n}Pln(P)$  with
$P=p^n$ going to $0$ when $p$ does.
using the well known limit
$\lim_{x\to 0^+}xln(x)=0$ we the desired result, which is $0$.

