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如图。
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x->0
sinx = x-(1/6)x^3 +o(x^3)
e^(sinx) = e^[x-(1/6)x^3 +o(x^3)]
tanx =x+(1/3)x^3 +o(x^3)
e^(tanx)= e^[ x+(1/3)x^3 +o(x^3) ]
e^(sinx) -e^(tanx)
=e^[x-(1/6)x^3 +o(x^3)] -e^[ x+(1/3)x^3 +o(x^3) ]
=e^x . { e^[-(1/6)x^3 +o(x^3)] -e^[(1/3)x^3 +o(x^3)] }
=e^x . [ 1 - (1/6)x^3 -1 - (1/3)x^3 +o(x^3) ]
=e^x .[ - (1/2)x^3 +o(x^3) ]
=- (1/2)x^3 +o(x^3)
lim(x->0) [e^(sinx) -e^(tanx)]/ [xln(1+x^2)]
=lim(x->0) -(1/2)x^3/ x^3
=-1/2
sinx = x-(1/6)x^3 +o(x^3)
e^(sinx) = e^[x-(1/6)x^3 +o(x^3)]
tanx =x+(1/3)x^3 +o(x^3)
e^(tanx)= e^[ x+(1/3)x^3 +o(x^3) ]
e^(sinx) -e^(tanx)
=e^[x-(1/6)x^3 +o(x^3)] -e^[ x+(1/3)x^3 +o(x^3) ]
=e^x . { e^[-(1/6)x^3 +o(x^3)] -e^[(1/3)x^3 +o(x^3)] }
=e^x . [ 1 - (1/6)x^3 -1 - (1/3)x^3 +o(x^3) ]
=e^x .[ - (1/2)x^3 +o(x^3) ]
=- (1/2)x^3 +o(x^3)
lim(x->0) [e^(sinx) -e^(tanx)]/ [xln(1+x^2)]
=lim(x->0) -(1/2)x^3/ x^3
=-1/2
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