
用洛必达法则求limx→0(1/x^2-cot^2x)的详细步骤
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limx→0(1/x^2-cot^2x)
=limx→0(1/x^2-sin^2x/cos^2x)
=limx→0[(sin^2x-x^2cos^2x)/x^2sin^2x]
=limx→0[(sin^2x-x^2)/x^2sin^2x]
=limx→0[(2sinxcosx-2x)/(2xsin^2x+2x^2sinxcosx)]
=limx→0[(2sinx-2x)/(2xsin^2x+2x^2sinx)]
=limx→0[(2cosx-2)/(2sin^2x+4xsinxcosx+2x^2cosx+4xsinx)]
=limx→0[(2cosx-2)/(2sin^2x+4xsinx+2x^2+4xsinx)]
=limx→0[-2sinx/(4sinxcosx+4sinx+4xcosx+4x+4sinx+4xcosx)]
=limx→0[-2sinx/(12sinx+12x)]
=limx→0[-2cosx/(12cosx+12)]
=-1/12
=limx→0(1/x^2-sin^2x/cos^2x)
=limx→0[(sin^2x-x^2cos^2x)/x^2sin^2x]
=limx→0[(sin^2x-x^2)/x^2sin^2x]
=limx→0[(2sinxcosx-2x)/(2xsin^2x+2x^2sinxcosx)]
=limx→0[(2sinx-2x)/(2xsin^2x+2x^2sinx)]
=limx→0[(2cosx-2)/(2sin^2x+4xsinxcosx+2x^2cosx+4xsinx)]
=limx→0[(2cosx-2)/(2sin^2x+4xsinx+2x^2+4xsinx)]
=limx→0[-2sinx/(4sinxcosx+4sinx+4xcosx+4x+4sinx+4xcosx)]
=limx→0[-2sinx/(12sinx+12x)]
=limx→0[-2cosx/(12cosx+12)]
=-1/12

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