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Find the limit

$$\lim_{x \rightarrow 1}(\log(ex))^{\frac{1}{\log x}}$$

I have the solution of this which goes like: \begin{align} \lim_{x \rightarrow 1}(\log(ex))^{\frac{1}{\log x}} & = \lim_{x \rightarrow 1} \left(\log(e) + \log(x) \right)^{\frac{1}{\log x}}\\ & = \lim_{x \rightarrow 1} \left(1 + \log(x) \right)^{\frac{1}{\log x}}\\ & = e^{\lim_{x \to 1} \dfrac{\log(x)}{\log(x)}}\\ & = e \end{align} I need the derivation of the property used in this solution. If somebody can detail me on this I will be greatful or let me know the source from where I can get derivation of results.

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up vote 3 down vote accepted

Note that the first step uses the property that $$\log(ab) = \log(a) + \log(b)$$ The second step makes use of the fact that $$\log(e) = 1$$ For the third step, note the following. Set $\log(x) = t$. Note that as $x \to 1$, we have $t \to 0$. Hence, the limit is $$\lim_{t \to 0}(1+t)^{1/t}$$ which I trust you should be able to evaluate.

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