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

Question

如题所述

Answer 1

lim<x→∞>[(x+2c)/(x-1)]^x=e^3
===> lim<x→∞>[(x-1+2c+1)/(x-1)]^x=e^3
===> lim<x→∞>[1+(2c+1/x-1)]^x=e^3
===> lim<x→∞>{[1+(2c+1/x-1)]^(x-1/2c+1)}^[(2c+1)x/(x-1)]=e^3
===> e^lim<x→∞>[(2c+1)x/(x-1)]=e^3
===> lim<x→∞>(2c+1)x/(x-1)=3
===> lim<x→∞>(2c+1)/[1-(1/x)]=3
===> (2c+1)/(1-0)=3
===> c=1

Answer 2

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

Answer 3