Evaluate the limit $\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$ I have this assignment 
Evaluate the limit:
5.$\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$
I don't think we are allowed to use L'Hopital, but I can't imagine how else. 

Let $t=-\ln x$. Then $x=e^{-t}$, and hence $$\lim_{x\to 0^+}x\ln x=\lim_{t\to +\infty}-e^{-t}t=\lim_{t\to +\infty}\frac{-t}{e^t}=0$$
