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$\begin{array}{lll}\lim_{x \rightarrow 0} x*\log(x) & = & \lim_{x\rightarrow 0} \frac{\log(x)}{x^{-1}} \\ { } & = & \lim_{x \rightarrow 0} \frac{x^{-1}}{-x^{-2}} \\ {} & = & \lim_{x\rightarrow 0}-x \\ { } & = & 0\end{array}$

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$3. \displaystyle{\lim_{x \to 0} \frac{e^x- 1}{x} = 1} \\[5mm] 4. \displaystyle{\lim_{x\ to 0} \frac{\log (1 + x)}{x} = 1} \\[5mm] 5. \displaystyle{\lim_{x \to \infty} \left( 1 + \dfrac{1}{x} \right)^n = e} \\[5mm] 6. \displaystyle{\lim_{x \to 0}(1 + x)^\frac{1}{x} = e} \\[5mm] 7. \displaystyle{\lim_{x \to 0}\frac{\tan x}{x} = 1}$

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