lim(1+tanx/1+sinx)^1/sinx x趋近于0
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lim(1+tanx/1+sinx)^1/sinx x趋近于0
![lim(1+tanx/1+sinx)^1/sinx x趋近于0](/uploads/image/z/15762483-27-3.jpg?t=lim%281%2Btanx%2F1%2Bsinx%29%5E1%2Fsinx+x%E8%B6%8B%E8%BF%91%E4%BA%8E0)
原式=lim(x->0){[1+(tanx-sinx)/(1+sinx)]^[((1+sinx)/(tanx-sinx))*((tanx-sinx)/(sinx+sin²x))]}
=e^{lim(x->0)[(tanx-sinx)/(sinx+sin²x)]} (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=e^{lim(x->0)[(1/cosx-1)/(1+sinx)]}
=e^[(1-1)/(1+0)]
=e^0
=1.
=e^{lim(x->0)[(tanx-sinx)/(sinx+sin²x)]} (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=e^{lim(x->0)[(1/cosx-1)/(1+sinx)]}
=e^[(1-1)/(1+0)]
=e^0
=1.
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