求助求助求助

x->0
分子
sinx=x-(1/6)x^3+o(x^4)
cos(sinx)
=1-(1/2)[x-(1/6)x^3]^2+ (1/24)[x-(1/6)x^3]^4+o(x^4)
=1-(1/2)[x^2-(1/3)x^4+...]+ (1/24)[x^4+...]+o(x^4)
=1-(1/2)x^4+(5/24)x^4+o(x^4)
cosx -1 = -(1/2)x^2 +(1/24)x^4+o(x^4)
e^(cosx -1)
=1 +[-(1/2)x^2 +(1/24)x^4] +(1/2)[-(1/2)x^2 +(1/24)x^4]^2 +o(x^4)
=1 +[-(1/2)x^2 +(1/24)x^4] +(1/2)[(1/4)x^4+...] +o(x^4)
=1-(1/2)x^2 + (1/6)x^4+o(x^4)
cos(sinx)-e^(cosx -1) =(5/24-1/6)x^4+o(x^4) = (1/24)x^4+o(x^4)
分母
(tanx)^2-(sinx)^2
=(tanx-sinx)(tanx+sinx)
~[(1/2)x^3](2x)
=x^4
//
lim(x->0) [cos(sinx)-e^(cosx -1)]/[(tanx)^2-(sinx)^2]
=lim(x->0) [(1/24)x^4]/[x^4]
=1/24