Question

lim x→0 1/x³[(2+cosx/3)-1]

求极限lim(x->0) 1/x^3 {[(2+cosx)/3]^x-1}
求极限lim(x->0)
1/x^3 {[(2+cosx)/3]^x-1}

Answer 1

=1/x^3{e^[ln((2+cosx)-ln3)*x]-1} e^x~1+x
=1/x^3[ln((2+cosx)-ln3)*x] ln(1+x)~x
=1/x^2[(2+cosx)/3-1]
=1/x^2[(cosx-1)/3]
=1/x^2(-2sin^2(x/2))/3
=-2/3*(1/2)^2
=-1/6