lim x-->pi^- ln(sin x) (i.e., the question states it aproaches pi from the left)?

Question

lim x-->pi^- ln(sin x)
(i.e., the question states it aproaches pi from the left)

If I start inside the parenthesis, as x aproaches pi (from the left) for sin(x) this part aproaches 0.

The answer in the back of the book says the correct answer is -INFINITY (for descriptive purposes).

I'm not sure what to do after that, Is the answer:
I'm guessing to drop the limit part and say ln(0) = -INFINITY (for descriptive purposes) because it arrives at the correct answer.

But I am concerned that I may be reaching the right answer for the wrong reasons. Is my method correct for reaching this answer? Is it right to start in the parenthesis part, and then work out?

If a similar question showed up on a test, how would I prove it. Puggy's answer has convinced me that it is correc and given some insight, but I think I need a proof if I got the question on a test.

Answer 1

ln(sin x)
lim x->pi_
As x gets closer and closer to pi, ln(sinx) goes very slowly towards - infinity. Whe x = pi ln(sinx) is undefined until x >2pi. Then the cycle repeats.
It is correct to say ln x approaches - infinity as x approaches o, but it is not correct to say ln(0) = - infinity. Rather, ln(0) is undefined.

Answer 2

lim ln(sin(x))
x -> pi-

One-sided limits are always tricky to evaluate. If you know the function is going to be undefined after plugging in the value of the limit, all you have to do is test a value close to the limit from the left; this would mean testing a value like x = 3.13, or, if you prefer, you can move closer like x = 3.135.

One-sided limits at functions not defined at a point will almost always be negative infinity or positive infinity. As long as you test a single point and get a large positive (or large negative) number, your answer is going to be negative or positive infinity, corresponding to the sign of that value.