Question

lim(x→0)（sinx/x）^(1/1-cosx)等于

Answer 1

lim(sinx/x)^[1/(1-cosx)]
x→0

= lime^[ln(sinx/x)]/(1-cosx)
x→0

= e^lim[ln(sinx/x)]/(1-cosx) (0/0)
x→0

= e^lim[ln(sinx/x)]/(x²/2)
x→0

= e^lim[(x/sinx)(xcosx-sinx)/x²]/x
x→0

= e^lim[(xcosx-sinx)/(x³)
x→0

= e^lim[(cosx-xsinx-cosx)/(3x^2)
x→0

= 1/e^(1/3)

=1/³√e

Answer 2

lim<x→0>(sinx/x)^[1/(1-cosx)]
= lim<x→0>e^[ln(sinx/x)]/(1-cosx)
= e^lim<x→0>[ln(sinx/x)]/(1-cosx) (0/0)
= e^lim<x→0>[(x/sinx)(xcosx-sinx)/x^2]/sinx
= e^lim<x→0>[(xcosx-sinx)/(x^2sinx)
= e^lim<x→0>[(xcosx-sinx)/(x^3) (0/0)
= e^lim<x→0>[(cosx-xsinx-cosx)/(3x^2) = e^(-1/3) = 1/e^(1/3)

Answer 3

1-cosx约等于1/2x平方