lim(x→∞)[x/(x-c)]的x次方=2,求Clim(x→∞)[x/(x-c)]的x次方=2,求C

Question

lim(x→∞) [x/(x-c)]的x次方=2,求C
lim(x→∞) [x/(x-c)] 的x次方=2,求C

Answer 1

lim [x/(x-c)]^x
x→∞
=lim [(x-c+c)/(x-c)]^x
x→∞
=lim [1 + c/(x-c)]^x
x→∞
=lim [1 + c/(x-c)]^{[(x-c)/c]×[cx/(x-c)]}
x→∞
=lim e^[cx/(x-c)]
x→∞
=lim e^[c/(1-c/x)]
x→∞
= e^c
令 e^c = 2
∴ c = ln2