高数lim【(sinx—siny)除以sin(x-y)】,x趋向于y,y不等于0
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高数lim【(sinx—siny)除以sin(x-y)】,x趋向于y,y不等于0
![高数lim【(sinx—siny)除以sin(x-y)】,x趋向于y,y不等于0](/uploads/image/z/16245310-22-0.jpg?t=%E9%AB%98%E6%95%B0lim%E3%80%90%28sinx%E2%80%94siny%29%E9%99%A4%E4%BB%A5sin%28x-y%29%E3%80%91%2Cx%E8%B6%8B%E5%90%91%E4%BA%8Ey%2Cy%E4%B8%8D%E7%AD%89%E4%BA%8E0)
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
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
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证明sin(x+y)sin(x-y)=(sinx)^2-(siny)^2.
证明:对x不等于y,恒有|sinx-siny|< |x-y|