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

Question

Is it true that:

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

Answer 1

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

Answer 2

Yes. Use L'Hospital:

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