用洛必达法则,x趋于派/2,求tanx/tan3x的极限;x趋于正的0时,求ln cotx/lnx的极限.

1.
用罗比达法则,求导
lim tanx/tan3x
=lim ( 1/cosx^2) / (3/cos3x^2)
= lim (cos3x^2)/3cosx^2
= lim 6cos3x(-sin3x)/6cosx(-sinx)
= lim sin6x/sin2x
= lim 6cos6x/2cos2x
= -6 / -2 = 3
2.
由洛比达法则
lim lncotx/lnx
=lim -x/(sin^2xcotx)
由等价无穷小x~sinx
上式=lim -tanx/sinx=lim -1/cosx=-1