高数菜鸟 x趋于无穷 lim[(x+2c)/(x-1)]^x=e^3 求c值

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高数菜鸟 x趋于无穷 lim[(x+2c)/(x-1)]^x=e^3 求c值
高数菜鸟 x趋于无穷 lim[(x+2c)/(x-1)]^x=e^3 求c值

高数菜鸟 x趋于无穷 lim[(x+2c)/(x-1)]^x=e^3 求c值
lim[(x+2c)/(x-1)]^x=e^3
===> lim[(x-1+2c+1)/(x-1)]^x=e^3
===> lim[1+(2c+1/x-1)]^x=e^3
===> lim{[1+(2c+1/x-1)]^(x-1/2c+1)}^[(2c+1)x/(x-1)]=e^3
===> e^lim[(2c+1)x/(x-1)]=e^3
===> lim(2c+1)x/(x-1)=3
===> lim(2c+1)/[1-(1/x)]=3
===> (2c+1)/(1-0)=3
===> c=1

t=x-1
lim((t+2c+1)/t)^(t+1)=e^3
lim(1+(2c+1)/t)^t=e^3
2c+1=3
c=1

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