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in this question, can I just do direct substitution? $$\lim_{x\to 0^+}\ln(\sin(x))$$


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No. What is $\ln(0)$? – David Mitra Sep 30 '12 at 22:00
oh yea its DNE, i see... so what is the correct step of solving it? – John Lee Sep 30 '12 at 22:03
up vote 3 down vote accepted

You can’t substitute $x=0$, since $\ln 0$ is undefined, but you can use the fact that $\sin x\to 0^+$ as $x\to 0^+$ to say that

$$\lim_{x\to 0^+}\ln\sin x=\lim_{x\to 0^+}\ln x\;.$$

That’s a limit that you should know: $$\lim_{x\to 0^+}\ln x=-\infty\;.$$

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That works when the function is continuous at the limit point. As it stands, the function is not (since it isn't defined there), so I wouldn't look at it that way. However you can substitute $u= \sin x$. Then as $x\to 0^+, u\to 0^+$, so

$$\lim_{x\to 0^+}\ln(\sin x)=\lim_{u\to 0^+}\ln u$$

The limit diverges to $-\infty$. It seems like you're trying to say that $\ln(0)=-\infty$, which is incorrect.

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$\ln( \lim_{x\to 0^+} \sin(x) )$

So as $x \to 0^+$, $\ln(\sin(x) ) \to -\infty$

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