lim(x->∞)[(x+2)/(x-1)]^mx=e^(-4) 则m

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lim(x->∞)[(x+2)/(x-1)]^mx=e^(-4) 则m

lim(x->∞)[(x+2)/(x-1)]^mx=e^(-4) 则m
lim(x->∞)[(x+2)/(x-1)]^mx=e^(-4) 则m

lim(x->∞)[(x+2)/(x-1)]^mx=e^(-4) 则m
[(x+2)/(x-1)]^mx
=[(x-1+3)/(x-1)]^mx
=[1+3/(x-1)]^mx
令a=(x-1)/3
x=3a+1
[1+3/(x-1)]^mx
=(1+1/a)^m(3a+1)
=(1+1/a)^3ma*(1+1/a)
=[(1+1/a)^a]^(3m)*(1+1/a)
x趋于0则a趋于0,所以1+1/a极限是1
[(1+1/a)^a]^(3m)极限=e^(3m)
所以3m=-4
m=-4/3

因为lim(x->∞)（1+1/x）^x=e^x
所以lim(x->∞)（[(x+2)/(x-1)]^mx=lim(x->∞)（[1+3/（x-1）]^[(x-1)/3 *3mx/(x-1)]=lim(x->∞)（e^[3mx/(x-1）]
因为lim(x->∞)（x/(x-1))=1
所以3m=-4

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