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

[oahsen]

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

 

[StatusX]

What exactly went wrong when you tried that?

 

[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

 

[dextercioby]

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

Daniel.

 

[marlon]

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

marlon

 

[dextercioby]

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

Daniel.

 

[marlon]

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

 

[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.