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I'm doing some practice questions and I've encountered a wall. The question is:

Find the limit of the function $(\ln4x-\ln(x+7))$ as $x \rightarrow \infty$.

the indeterminate form is $\infty-\infty$, so I'm not sure how to advance from here. I think I would like $f(x)/g(x)$ form, but I'm not sure how to do that in this situation. Can you give me a hint in the right direction?

I certainly don't want the answer.

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  • $\begingroup$ What is $\ln(a)-\ln(b)$? $\endgroup$ – Kitegi Mar 29 '15 at 22:24
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HINT: $$\ln4x -\ln{(x+7)}=\ln\frac{4x}{ x+7}$$ from the properties of the logarithm.

Evaluate $\displaystyle \frac{4x}{x+7}$ and take the logarithm of your answer.

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  • $\begingroup$ Actually it's $ln ( \frac{4x}{x+7} ). $\endgroup$ – Patrick Mar 29 '15 at 22:21
  • $\begingroup$ i.e. the log of the entire expression. $\endgroup$ – Patrick Mar 29 '15 at 22:21
  • $\begingroup$ Apologies edited now. Can we delete these comments? $\endgroup$ – Permian Mar 29 '15 at 22:26
  • $\begingroup$ I got the right answer. If you didn't answer, I wouldn't have known to evaluate the inside of the log first. Thank you. $\endgroup$ – NewbieNick Mar 31 '15 at 2:47
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Hint: $$\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)$$

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    $\begingroup$ Don't forget to use \ln instead of ln to produce the proper non-italic text, e.g. $\ln$ vs $ln$. $\endgroup$ – MathMajor Mar 29 '15 at 22:30
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Hint

Apply properties of logarithms to get

$$\ln 4x-\ln (x+7)=\ln \frac{4x}{x+7}.$$

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