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I have to solve: $\lim\limits_{x\to \infty} \frac{\ln(x+1)}{\ln(x)}$. Can you give me any hints to go? Thanks a lot!

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up vote 3 down vote accepted

$$\lim\limits_{x\to +\infty} \frac{\ln\left(x+1 \right)}{\ln(x)}=\lim\limits_{x\to +\infty} \frac{\ln\left(x\left(1+\frac{1}{x}\right)\right)}{\ln(x)}=\lim\limits_{x\to +\infty} \frac{\ln x+\ln \left(1+\frac{1}{x}\right)}{\ln(x)}=1$$

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L’Hospital’s rule is a perfectly good, straightforward way to evaluate the limit, and in this case it’s easy; there’s no reason not to use it. However, there is also a pretty simple way to get it more directly. Start by rewriting the numerator:

$$\ln(x+1)=\ln x\left(1+\frac1x\right)=\ln x+\ln\left(1+\frac1x\right)\;.$$

I’ll spoiler-protect the rest to give you a chance to work it out if you want; mouse-over to see it.

$\displaystyle\frac{\ln(x+1)}{\ln x}=\frac{\ln x+\ln\left(1+\frac1x\right)}{\ln x}=1+\frac{\ln\left(1+\frac1x\right)}{\ln x}$, whose limit as $x\to\infty$ is easy to evaluate.

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hint:use L'Hôpital's rule to solve it

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Use the dominant terms in numerator and denominator and you're done.

$$\lim_{x\to \infty} \ln(x+1)/\ln(x)=\lim_{x\to \infty} \ln(x)/\ln(x)=1$$

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the first equality is not clear, can you explian ? – Belgi Sep 15 '12 at 10:57
What justifies the first equality? – Pedro Tamaroff Sep 15 '12 at 22:13
@Peter Tamaroff: my sister used a comment above. (Chris) – user 1618033 Sep 16 '12 at 12:09
As belgi is stating the first step is not clear – Pedro Tamaroff Sep 16 '12 at 15:04

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