Why is the limit of ln(x)/arctan(x) as x approaches 0 from the right, not zero?

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Homework Help Overview

The discussion revolves around evaluating the limit of the function ln(x)/arctan(x) as x approaches 0 from the right. Participants are exploring why this limit is not zero, despite initial attempts suggesting otherwise.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the application of L'Hopital's rule and question its validity in this context. There are attempts to evaluate the limit by substituting small values of x and analyzing the behavior of the function near zero.

Discussion Status

The discussion is ongoing, with some participants providing insights into the behavior of the function as x approaches 0. There is a recognition that the limit does not approach zero, and the implications of this observation are being explored.

Contextual Notes

Some participants note that the function exhibits unbounded behavior as x approaches 0, raising questions about the formulation of the original problem.

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Homework Statement



Why is the limit of ln(x)/arctan(x) as x approaches 0 from the right, not zero?

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The Attempt at a Solution



I used L'hopital's rule and got zero. But question specifically states that is not the answer.
 
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$$\lim_{x \rightarrow 0^+} \ln(x) = -\infty$$
whereas
$$\lim_{x \rightarrow 0^+} \arctan(x) = 0$$
so L'Hopital's rule does not apply.
 
As jbunniii said, L'Hopital's rule does not apply here. Notice that If x= .000001, we have
\frac{ln(.000001)}{arctan(.000001)}= \frac{-13.8155}{.000001}= -13815510
not anywhere near 0!
 
Thank you!

But, that's all there is to it? I don't have to simplify anything? Just plug in values?
 
No, I didn't say that. In order to prove that a limit is a specific number you have to prove that it gets arbitrarily close to that number. My point was that for x very close to 0, the function value is very far away from 0. It is theoretically possible that a function would turn from being around -13 million at x= .000001 to 0 at x= 0, but that would be a very strange function!
 
It's so obviously unbounded that it's rather a strange question. Are you sure you've stated it correctly?
 

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