Computing $\lim\limits_{x \to \infty } \frac{x}{\ln\big(\! \large\frac{2^{x}}{x}\! \big)}$

Can I use the L'Hopital rule in order to find $$\lim_{x \to \infty } \frac{x}{\ln\left ( \large\frac{2^{x}}{x} \right )} \qquad ?$$

Thank you!

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$$\frac{x}{\log\frac{2^x}{x}}=\frac{x}{x\log 2-\log x}=\frac{1}{\log 2-\frac{\log x}{x}}$$

But using L'Hospital, we get

$$\lim_{x\to\infty}\frac{\log x}{x}=\lim_{x\to\infty}\frac{1}{x}=0$$

so using this and arithmetic of limits in the first part, we finally get

$$\lim_{x\to\infty}\frac{x}{\log\frac{2^x}{x}}=\frac{1}{\log 2-0}=\frac{1}{\log 2}$$

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If you're going to use L'Hospital, why not just use it on $$\frac{x}{x\log(2)-\log(x)}$$ ? +1 in any case :-) –  robjohn Feb 21 '13 at 13:29
Because, and this is not a rude answer, I don't want to...:) In fact, I think that the little algebraic manipulation shown above renders an easier and clearer path to the result. We could as well have used L'H from the very beginning. –  DonAntonio Feb 21 '13 at 13:31
Fair enough. Your answer is quite fine; I just went for the quick kill. As I was mentioning in another comment, $\lim\limits_{x\to\infty}\frac{\log(x)}{x}=0$ is a useful limit to know. –  robjohn Feb 21 '13 at 13:35

Yes you can. It will be slightly easier if you first use $$\ln(2^x/x)=\ln(2)x-\ln(x)$$ to simplify the expression.

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@Tania: Note that $$\frac{a}{b}=\frac{1}{\frac{b}{a}},~~a\neq0$$. –  B. S. Feb 21 '13 at 12:48
@ Michalis Why from there I don't really need to use l'Hospital anymore? Thank you –  Tania Feb 21 '13 at 13:12
@Tania: Until you are comfortable using $\lim\limits_{x\to\infty}\frac{\log(x)}{x}=0$, you should use L'Hospital. –  robjohn Feb 21 '13 at 13:27
@Tania: Robjohn is right. Using l'Hospital is the easiest way to proceed, so I removed my comment in the end. –  Michalis Feb 21 '13 at 13:38
This limit can be evaluated by use of the Hopital theorem, but to be correct you should first of all show that $lim(x\rightarrow \infty)$ $ln(2^x/x)=\infty$. Only now you can be sure that the Hopital hypothesis are verified. You should though observe that the use of Hopital in this case is rather excessive since you have to evaluate the derivative of a fraction. It is always a good excercise with Hopital to ask yourself if you cannot adopt some other method.