What is $\lim_{x \rightarrow 0} x\log(x)$?

Question

Is it true that:

$$\lim_{x \rightarrow 0} x\log_{n}(x) = 0 \quad ; \quad n \in \mathbb{R}$$

@Arjang Since $\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$ it doesn't matter for this question. — sTertooy, Jan 08 '17 at 01:46
I think in US $\log=\ln$ when the base is not specified. At least this is what it means in all software that I know. — Momo, Jan 08 '17 at 01:51

score 3 · Answer 1 · answered Jan 08 '17 at 01:44

3

Yes. Use L'Hospital:

$$\lim_{x\rightarrow 0}x\log x=\lim_{x\rightarrow 0}\frac{\log x}{1/x}=\lim_{x\rightarrow 0}\frac{1/x}{-1/x^2}=\lim_{x\rightarrow 0}-x=0$$

answered Jan 08 '17 at 01:44

Momo

score -1 · Answer 2 · answered Jan 08 '17 at 01:43

-1

hint: Write $x = \dfrac{1}{\frac{1}{x}}$

answered Jan 08 '17 at 01:43

DeepSea

2 Answers2