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Is it true that:

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

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    is this $\log_{10} x$ or $\log_{e} x$ – 2017-01-08
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    This should be in your calculus textbook. – 2017-01-08
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    @Arjang Since $\log_{10}(x) = \frac{\ln(x)}{\ln(10)}$ it doesn't matter for this question. – 2017-01-08
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    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. – 2017-01-08

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Yes. Use L'Hospital:

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

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

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    How does that help? – 2017-01-08