lim(1+tanx/1+sinx)^1/sinx x趋近于0

lim(1+tanx/1+sinx)^1/sinx x趋近于0
lim(1+tanx/1+sinx)^1/sinx x趋近于0

lim(1+tanx/1+sinx)^1/sinx x趋近于0
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.