# Limit $\lim_{x\to \infty } \, \frac{x}{\ln (x)-\ln \left(\frac{1}{x}\right)}$

Wolfram Alpha evaluates this limit

$$\lim_{x\to \infty } \, \frac{x}{\ln (x)-\ln \left(\frac{1}{x}\right)}$$

to be infinity.

But I suspect it could be a real number. What is the correct answer?

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Can someone fix the title? –  wj32 Oct 27 '12 at 8:14
Why do you suspect it could be a real number? –  Chris Eagle Oct 27 '12 at 10:16

Wolfram alpha is right because the denominator is $\ln(x)-\ln(\frac1x)=2\ln x$, which grows slower than $x$.