lim(x->∞)[(x+2)/(x-1)]^mx=e^(-4) 则m
问题描述：
lim(x->∞)[(x+2)/(x-1)]^mx=e^(-4) 则m
答
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
答
[(x+2)/(x-1)]^mx=[(x-1+3)/(x-1)]^mx=[1+3/(x-1)]^mx令a=(x-1)/3x=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)...