解法一：重要极限limsinx/x=1
lim(sinx-sinα)/(x-α)
=lim2cos[(x+α)/2]sin[(x-α)/2]/(x-α)
=limcos[(x+α)/2]sin[(x-α)/2]/[(x-α)/2]
此处limsin[(x-α)/2]/[(x-α)/2]=1,等价无穷小代换
=limcos[(x+α)/2]
=cosα
解法二：罗必塔法则
lim(sinx-sinα)/(x-α)
0/0型未定式,分子、分母分别求导
=limcosx
=cosα