find the limit of (tg(x))^tg(2x) as x tends to zero

[dariosabino]

limit of (tg(x))^tg(2x) as x tends to zero.

 

[stapel]

limit of (tg(x))^tg(2x) as x tends to zero.

Does "tg" mean "tan" for "tangent"? Logarithmic differentiation looks to be useful for this exercise. Have you started with that method, or are you proceeding by another route? How far have you gotten? Where are you stuck?

Please be complete. Thank you!

 

[dariosabino]

yes, tangent!!! i guess it would be useful to do this

exp lim ln(tan(x))/(1/tan(2x))

but i´m stucked here

 

[stapel]

i guess it would be useful to do this

exp lim ln(tan(x))/(1/tan(2x))

Instead of exponentiating, try using logarithms. In particular:

. . . . .\(\displaystyle y\, =\, (\tan(x))^{\tan(2x)}\)

. . . . .\(\displaystyle \ln(y)\, =\, \tan(2x)\, \ln\left(\tan(2x)\right)\, = \, \frac{\ln\left(\tan(2x)\right)}{\left(\frac{1}{tan(x)}\right)}\)

Then differentiate, etc. I haven't chased it all down, but it looks like it may get messy. Make sure you end up with the limit of the log of y being zero, so that the limit of y is 1.