x->0
分母
1-cosx ~ (1/2)x^2
x(1-cosx) ~ (1/2)x^3
分子
e^x ~ 1+x + (1/2)x^2 + (1/6)x^3
e^(tanx)
~e^(x+(1/3)x^3)
~ 1+(x+(1/3)x^3) + (1/2)[(x+(1/3)x^3)]^2 + (1/6)[(x+(1/3)x^3)]^3
~ 1+x +(1/2)x^2 +(1/2)x^3
e^x -e^(tanx) ~(1/6 - 1/2)x^3 = -(1/3)x^3
//
lim(x→0) [e^x-e^(tanx) ]/[x(1-cosx)]
=lim(x→0) -(1/3)x^3/[ (1/2)x^3]
=-2/3
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