高等数学吧 关注：472,149贴子：3,879,517
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sinx = x-(1/6)x^3 +o(x^3)
sin(sinx)
=[x-(1/6)x^3]-(1/6)[x-(1/6)x^3]^3 +o(x^3)
=[x-(1/6)x^3]-(1/6)[x^3+...] +o(x^3)
=x-(1/3)x^3 +o(x^3)
arctanx = x -(1/3)x^3 +o(x^3)
sin[arctanx]
=[x -(1/3)x^3]-(1/6)[x -(1/3)x^3]^3 +o(x^3)
=[x -(1/3)x^3]-(1/6)[x^3+...] +o(x^3)
=x - (1/2)x^3 +o(x^3)
sin(sinx)/sin[arctanx]
~ [x-(1/3)x^3]/[x - (1/2)x^3]
= 1 + (1/6)x^3/[x - (1/2)x^3]
~ 1 + (1/6)x^2 +o(x^2)
lim(x->0) { sin(sinx)/sin[arctanx] }^[1/(1-cosx)]
=lim(x->0) { sin(sinx)/sin[arctanx] }^(2/x^2)
=lim(x->0) [ 1 + (1/6)x^2 ]^(2/x^2)
=e^(1/3)
sin(sinx)
=[x-(1/6)x^3]-(1/6)[x-(1/6)x^3]^3 +o(x^3)
=[x-(1/6)x^3]-(1/6)[x^3+...] +o(x^3)
=x-(1/3)x^3 +o(x^3)
arctanx = x -(1/3)x^3 +o(x^3)
sin[arctanx]
=[x -(1/3)x^3]-(1/6)[x -(1/3)x^3]^3 +o(x^3)
=[x -(1/3)x^3]-(1/6)[x^3+...] +o(x^3)
=x - (1/2)x^3 +o(x^3)
sin(sinx)/sin[arctanx]
~ [x-(1/3)x^3]/[x - (1/2)x^3]
= 1 + (1/6)x^3/[x - (1/2)x^3]
~ 1 + (1/6)x^2 +o(x^2)
lim(x->0) { sin(sinx)/sin[arctanx] }^[1/(1-cosx)]
=lim(x->0) { sin(sinx)/sin[arctanx] }^(2/x^2)
=lim(x->0) [ 1 + (1/6)x^2 ]^(2/x^2)
=e^(1/3)
