A limit problem

1. Dec 21, 2006

oahsen

1. The problem statement, all variables and given/known data
f(x)=(tan(x)/x)^(1/(x^2)) it asks the limit of this function when x goes to 0

2. Relevant equations

3. 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

2. Dec 21, 2006

StatusX

What exactly went wrong when you tried that?

3. Dec 21, 2006

marlon

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: Dec 21, 2006
4. Dec 21, 2006

dextercioby

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

Daniel.

5. Dec 21, 2006

marlon

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

marlon

6. Dec 21, 2006

dextercioby

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

Daniel.

7. Dec 21, 2006

marlon

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: Dec 21, 2006
8. Dec 21, 2006

dextercioby

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.