Limiting f(x): Finding the Limit of (tan(x)/x)^(1/(x^2))

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Homework Help Overview

The problem involves finding the limit of the function f(x) = (tan(x)/x)^(1/(x^2)) as x approaches 0. The context is centered around limits and the behavior of trigonometric functions near zero.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss using the natural logarithm and L'Hôpital's rule, with one participant suggesting a Taylor series expansion for tan(x). There are questions about the validity of these approaches and the implications of the limit being non-zero.

Discussion Status

The discussion is ongoing, with various methods being explored. Some participants express uncertainty about the results, while others indicate potential errors in their calculations. There is no clear consensus on the limit's value, and multiple interpretations are being considered.

Contextual Notes

Participants note that the Taylor series for tan(x) is valid for |x| < pi/2, and there are concerns about the limit approaching infinity depending on the direction from which x approaches 0. There is also mention of the impact of assumptions on the outcome of the limit.

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Homework Statement


f(x)=(tan(x)/x)^(1/(x^2)) it asks the limit of this function when x goes to 0


Homework Equations





The Attempt at a Solution



i have tried to take the ln of the two sides than used the l'hopital rule but with that way i could not reach anything. pls help me
 
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What exactly went wrong when you tried that?
 
Develop the tan(x) in a series first. Using the actual tan(x) always gives me infinity divided by zero.

The Taylor series of tan(x) around zero is valid for |x| < pi/2 so...


marlon
 
Last edited:
If the limit is NOT 0, then Marlon's suggested method leads to an erroneous result.

Daniel.
 
dextercioby said:
If the limit is NOT 0, then Marlon's suggested method leads to an erroneous result.

Daniel.

Actually, YES, you are right. Actually, i don't know how to solve it so i am going to say it's indefinite :smile:

marlon
 
I get infinity, plus or minus, depending on whether the limit is approaching 0 from below or from above.

Daniel.
 
dextercioby said:
I get infinity, plus or minus, depending on whether the limit is approaching 0 from below or from above.

Daniel.

Yeah, (1+x^2)^(1/x^2) for x--> 0 (after doing the Taylor thing) gives me this : 1 + x > 1 and the power gets bigger if x gets towards 0, so you are EVOLVING towards infinity but what i cannot achieve is prove that the value is actually infinite

Also, if x is coming from te negative side, you are again evolving toward positive infinity because tanx/x = 1 + (x^2)/3 + ... with all positive powers !

marlon
 
Last edited:
I ****ed up the derivatives, I've had too much to drink at the party, apparently. By using the method suggested by Marlon, i now get e^{3}.

I won't go through that l'Hopital again. :d

Daniel.
 

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