Compute $\lim\limits_{n\to \infty} \ln(3n+7) - \ln(n)$

Question

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.

Answer 1

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

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

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

Answer 2

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...

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