高数lim(sinx—siny)除以sin(x-y),x趋向于y,y不等于0

Question

高数lim【(sinx—siny)除以sin(x-y)】,x趋向于y,y不等于0

Answer 1

令t=x-y
原式=lim(x->y) 2cos[(x+y)/2]sin[(x-y)/2]/sin(x-y) （和差化积）
=lim(t->0) 2cos[(x+y)/2]sin(t/2)/sint
=lim(t->0) cos[(x+y)/2]/cos(t/2)
=cos[(x+y)/2]
=cosy