Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.