发疯的鱼
幼苗
共回答了24个问题采纳率:100% 举报
=lim[f(sin^2x+cosx)/x^2](等价无穷小的替换)
因f'(1)存在
则f(1+x)-f(1)=f'(1)x+o(x)
即f[1+(sin^2x+cosx-1)]=f'(1)(sin^2x+cosx-1)+o(sin^2x+cosx-1)
=f'(1)(sin^2x+cosx-1)+o(x^2)
从而lim[f(sin^2x+cosx)/x^2]
=lim{[f'(1)(sin^2x+cosx-1)+o(x^2)]/x^2}
=f'(1)lim[(sin^2x+cosx-1)/x^2]
=f'(1)lim[(2sinxcosx-sinx)/(2x)]
=1/2*f'(1)lim(2cosx-1)*lim(sinx/x)
=1/2*f'(1)
1年前
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