[Math] apply L’Hôpital to $\lim_{x \to \infty} \frac{x+\ln x}{x-\ln x}$


Can I apply L'Hôpital to this limit:
$$\lim_{x \to \infty} \frac{x+\ln x}{x-\ln x}?$$
I am not sure if I can because I learnt that I use L'Hôpital only if we have $\frac{0}{0}$ or $\frac{\infty}{\infty}$ and here $x-\ln x$ is $\infty-\infty$ and x tends to infinity.

Best Answer

As an alternative to Stefano's calculation, note that the derivative of $x-\log x$ is $1-\frac1x$, which is $\ge \frac12$ whenever $x\ge 2$. Thus, by the mean value theorem we have $$ x-\log x \ge 2-\log 2 + \frac{x-2}{2} $$ for all $x\ge 2$, and the right-hand side of this clearly goes to $\infty$.

So $x-\log x\to \infty$ when $x\to\infty$.

It is also abundantly clear that $x+\log x$ goes to $\infty$ for $x\to\infty$, so you're allowed to try using L'Hospital on your fraction.

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