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Suppose we want to evaluate

$$\lim_{x \to 0} \; x \log (x)$$

If we write this in the form $x/(1/\log(x))$, then l'Hôpital's rule does not work, but if we write it as $\log(x)/(1/x)$, it does. Is there any sort of general guideline to choosing the numerator and denominator?

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    Note that the limit as written does not exist, and L'Hopital's is irrelevant: $\log x$ is not defined on an interval containing $0$: it's only defined to the right of $0$. Of course, you can take the limit as $x\to 0^+$. Rule of thumb: make choices that make the derivatives *simpler* or at least *no harder* than the original. $\frac{1}{\log x}$ has a derivative which is harder than the original, so it's not a good choice.2011-09-08

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