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春芽
共回答了22个问题采纳率:90.9% 举报
从您的解题过程猜测原题可能是 x->0, lim[ 1/x^2 - (cotx)^2]
错误的原因是等价无穷小只能替换因式,你上面的替换分子就有问题了,分母的替换是正确的.
lim[ 1/x^2 - (cotx)^2]
=lim[ 1/x^2 - (cscx)^2] +1 (利用(cscx)^2 =(cotx)^2+1)
=lim[ (sinx)^2 - x^2] / [x^2* (sinx)^2] + 1 (通分)
=lim[ (sinx)^2 - x^2] / (x^4) + 1 (分母等价无穷小替换)
=lim[ 2sinxcosx - 2x] /(4x^3) + 1(洛必达法则)
=lim[ sin2x -2x] / (4x^3) + 1 (三角公式sin2x=2sinxcosx)
=lim[2cos2x - 2] /(12x^2) + 1(洛必达法则)
= lim(cos2x-1)/ (6x^2) +1
=lim(-2sin2x)/(12x) +1(洛必达法则,也可用等价无穷小)
=lim(-4x)/(12x) +1
=2/3
1年前
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