lim(1-x平方/2)的1/sinx平方次 x趋向于0
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lim(1-x平方/2)的1/sinx平方次 x趋向于0
=[x →0]lim(1-x²/2)^(1/sinx)
=[x →0]lime^[ln(1-x²/2)^(1/sinx)]
=[x →0]lime^[(1/sinx)*ln(1-x²/2)]
=[x →0]lime^[ln(1-x²/2)/sinx]
=[x →0]lime^[-x/(1-x²/2)/cosx]
=e^[-0/(1-0)/cos0]
=e^-0
=1
再问: 上面不是1/sinx 是1/sinx的平方。
再答: =[x →0]lim(1-x²/2)^(1/sinx)² =[x →0]lime^[ln(1-x²/2)^(1/sinx)²] =[x →0]lime^[(1/sinx)²*ln(1-x²/2)] =[x →0]lime^{[ln(1-x²/2)]/sin²x} =[x →0]lime^{[-x/(1-x²/2)]/[2sinxcosx]} =[x →0]lime^{-x/(1-x²/2)*(2sinxcosx)} =[x →0]lime^{-x/(2-x²)(sinxcosx)} =[x →0]lime^{(-1)/[-2x(sinxcosx)+(2-x²)(cos²x-sin²x)]} =e^{(-1)/[0+2(1)]} =e^{-1/2} =1/√e
=[x →0]lime^[ln(1-x²/2)^(1/sinx)]
=[x →0]lime^[(1/sinx)*ln(1-x²/2)]
=[x →0]lime^[ln(1-x²/2)/sinx]
=[x →0]lime^[-x/(1-x²/2)/cosx]
=e^[-0/(1-0)/cos0]
=e^-0
=1
再问: 上面不是1/sinx 是1/sinx的平方。
再答: =[x →0]lim(1-x²/2)^(1/sinx)² =[x →0]lime^[ln(1-x²/2)^(1/sinx)²] =[x →0]lime^[(1/sinx)²*ln(1-x²/2)] =[x →0]lime^{[ln(1-x²/2)]/sin²x} =[x →0]lime^{[-x/(1-x²/2)]/[2sinxcosx]} =[x →0]lime^{-x/(1-x²/2)*(2sinxcosx)} =[x →0]lime^{-x/(2-x²)(sinxcosx)} =[x →0]lime^{(-1)/[-2x(sinxcosx)+(2-x²)(cos²x-sin²x)]} =e^{(-1)/[0+2(1)]} =e^{-1/2} =1/√e
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