The reason why I'm having trouble with this problem is because it involves natural log (ln) and I need to find the limit.

I need to find $\lim_{n\to\infty} \ln(3n+7)-\ln(n)$.

I noticed that as $n$ approaches infinity, $-\ln(n)$ should be approaching $-\infty$ but I'm having trouble finding the limit since $\ln(3n+7)$ is in the sequence.


Hint: $$\ln(x) - \ln(y) = \ln(\frac{x}{y})$$

Edit: $$\ln(3n + 7) - \ln(n) = \ln(\frac{3n+7}{n}) = \ln(3 + \frac7n)$$

$$\text{If }\lim_{n\to\infty}\frac7n = 0\text{, what remains?}$$

  • $\begingroup$ So I rewrote the sequence as ln((3n+7)/n) but I'm still not sure what the limit would be. I plugged in numbers for n to find the first few terms and I was getting results closer to one but it appears that is not working. $\endgroup$ – JMartinez Feb 10 '15 at 3:26
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    $\begingroup$ Would it be that ln(3+0) is what remains since limn→∞7n=0? $\endgroup$ – JMartinez Feb 10 '15 at 3:36
  • $\begingroup$ @JMartinez, Exactly! Notice that it's near impossible to guess the convergence of a sequence based on any patterns in the first few terms. I think you'll find this function doesn't tend to $1$ $\endgroup$ – jameselmore Feb 10 '15 at 3:37
  • $\begingroup$ Yea I noticed that it doesn't tend to 1 but 1.098612289 $\endgroup$ – JMartinez Feb 10 '15 at 3:40
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    $\begingroup$ @JMartinez, could you please select the answer if you found it useful? $\endgroup$ – jameselmore Feb 10 '15 at 4:03

We have \begin{align} \lim_{n\rightarrow\infty}\left(\log\left|3n+7\right|-\log\left|n\right|\right),\tag{1} \end{align} and if you recall your rules of logarithms, which can be viewed here if you need a quick reference (I would highly recommend you take a look as Math Insight is a great site), and these allow us to re-write the above as \begin{align} \lim_{n\rightarrow\infty}\left(\log\left|\frac{3n+7}{n}\right|\right)=\lim_{n\rightarrow\infty}\left(\log\left|3+\frac{7}{n}\right|\right),\tag{2} \end{align} and I'm sure you can do the rest...

Note: I, personally, use $\log\left|x\right| \Longleftrightarrow \log_{e}\left|x\right|$, but you may also say $\text{ln}\left|x\right|$...


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