What is the limit as x approaches 0 from the positive side of (sin x)(ln x)?

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Homework Help Overview

The problem involves finding the limit as x approaches 0 from the positive side of the expression (sin x)(ln x). This falls under the subject area of calculus, specifically limits and indeterminate forms.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply L'Hôpital's rule after rewriting the limit expression, but expresses uncertainty about the validity of the approach due to the nature of the indeterminate form. Other participants suggest alternative forms and approaches, including different applications of L'Hôpital's rule and the use of limits involving sin(x).

Discussion Status

The discussion is ongoing, with participants exploring various approaches to the limit. Some guidance has been offered regarding the application of L'Hôpital's rule, but there is no explicit consensus on the best method to proceed. Participants are questioning the clarity of expressions and the correctness of steps taken in their calculations.

Contextual Notes

There is a noted concern about the interpretation of the limit as x approaches 0 from the positive side, particularly regarding the use of direct substitution and the nature of indeterminate forms. The original poster also mentions uncertainty about the application of L'Hôpital's rule in this context.

phil ess
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Homework Statement



Find the limit as x -> 0+ of (sin x)(ln x)

Homework Equations



None

The Attempt at a Solution



I rewrote this as (sin x) / (1/ln x), then using L'Hopital it becomes:

(cos x) / [(-ln x) / (ln x)2] = [(ln x)2 cos x] / (-ln x)

So I get limit as x -> 0+ of [(ln x)2 cos x] / (-ln x)

Which isn't any better. I'm not sure if I can even use L'Hopital here because it requires that substitution gives the indeterminate state, but I don't know if 0+/0+ really counts. Any thoughts?
 
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Try (cos x)/[(-1) / (xln x)2]
 
It's probably easier to try l'Hopital on ln(x)/(1/sin(x)). What does that give you? Remember lim sin(x)/x=1.
 
Doing it on ln(x)/(1/sin(x)) gives me sin(x)/-xcos(x). Now what do I do though? I can't solve using direct substitution because 0+ isn't really a number. Do I just substitute in something close like 0.0001?

I guess I could rewrite it -tan(x)/x but that doesn't help me much.
 
Use more parentheses! I can't tell what sin(x)/-xcos(x) is supposed to mean. The closest it could be to being right is -(sin(x)/x)*cos(x). But that's not even right, the power of the sin is wrong. Can you fix it?
 

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