x趋近正无穷 ((x^2-2x+1)/(x^2-4x+2))^x 的极限是
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x趋近正无穷 ((x^2-2x+1)/(x^2-4x+2))^x 的极限是
如题
如题
y = [(x^2 - 2x + 1)/(x^2 - 4x + 2)]^x
lny = x * [2ln(x-1)-ln(x^2-4x+2)] =[2ln(x-1)-ln(x^2-4x+2)]/x^-1
因为 lny 的极限属于 0 * ∞ 型,所以应用罗必塔法则,有:
lim(lny) = lim[2/(x-1) - 2(x-2)/(x^2-4x+2)]/(-x^-2)
= -2*lim[x/(x-1) - x*(x-2)/(x^2-4x+2)]/x^-1
= -2*lim[1+1/(x-1) - 1 - (2x-2)/(x^2-4x+2)]/x^-1
= -2*lim[1/(x-1) - 2(x-1)/(x^2-4x+2)]/x^-1
= -2*lim(x^2-4x+2 -2x^2 +4x -2)/[(x-1)(x^2-4x+2)*x^-1]
= -2*lim(-x^2)*x/[(x-1)(x^2-4x+2)]
= 2*lim{1/[(1-1/x)(1-4/x+2/x^2)]}
= 2*1/[(1-0)(1-4*0+2*0)]
= 2
所以,y = e^2
lny = x * [2ln(x-1)-ln(x^2-4x+2)] =[2ln(x-1)-ln(x^2-4x+2)]/x^-1
因为 lny 的极限属于 0 * ∞ 型,所以应用罗必塔法则,有:
lim(lny) = lim[2/(x-1) - 2(x-2)/(x^2-4x+2)]/(-x^-2)
= -2*lim[x/(x-1) - x*(x-2)/(x^2-4x+2)]/x^-1
= -2*lim[1+1/(x-1) - 1 - (2x-2)/(x^2-4x+2)]/x^-1
= -2*lim[1/(x-1) - 2(x-1)/(x^2-4x+2)]/x^-1
= -2*lim(x^2-4x+2 -2x^2 +4x -2)/[(x-1)(x^2-4x+2)*x^-1]
= -2*lim(-x^2)*x/[(x-1)(x^2-4x+2)]
= 2*lim{1/[(1-1/x)(1-4/x+2/x^2)]}
= 2*1/[(1-0)(1-4*0+2*0)]
= 2
所以,y = e^2
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