# Prove that the following limit exists and find it!

Prove that $\lim_{x\to0^{+}} (-x) \ln x$ exists and find it.

I know we should write $(-x)*\ln(x)$ as $\ln(x)/1/x$. Then we can see by L'Hopital's Rule that it equals $x$. And as $x$ approaces $0^{+}$ the limit equals $0$. However I am not sure how to prove the limit exists? Do we have to do a $\epsilon -\delta$ proof?

• Yeah it is equal to zero and I think that's a good proof. – Shahar May 11 '14 at 22:35
• If you can already do a proof using a theorem and still wants an even elementary proof, I would say refer how the theorem is proved and repeat the same step for this specific application of theorem. – jdoicj May 13 '14 at 17:08

$$0\ <\ \left\vert\,x\ln\left(x\right)\,\right\vert\ \leq\ \left\vert\,x\left(x - 1\right)\right\vert$$