x-0 lim(e^x-e^-x)/sinx
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x-0 lim(e^x-e^-x)/sinx
x-0 lim(e^x-e^-x)/sinx
x-0 lim(e^x-e^-x)/sinx
运用洛必达法则对分子分母同时求导
(e^x-e^-x)'=e^x+e^x=2e^x
(sinx)'=cosx
当x=0时,2e^x=2,cosx=1
所以x-0 lim(e^x-e^-x)/sinx=2/1=2
原式=lim(x->0) e^(-x)*[e^(2x)-1]/sinx
=lim(x->0) e^(-x)*2x/x
=2
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