一道极限题设f(x)在x=0附近连续,且lim(1+x+f(x)/x)^(1/x)=e^3(x趋于0),求lim(1+f(x)/x)^(1/x)(x趋于0).

Question

一道极限题
设f(x)在x=0附近连续,且lim(1+x+f(x)/x)^(1/x)=e^3(x趋于0),求lim(1+f(x)/x)^(1/x)(x趋于0).

Answer 1

lim (1+x)^x =e => lim (1+x/3)^(1/x) = e^3
lim(1+x+f(x)/x)^x = e^3 ==> 1+x+f(x)/x (1+x/3) => f(x)~- 2x^2/3
所以 lim (1+f(x)/x)^(1/x) (1+(-2x/3)) ^(1/x)
令 t= -2x/3,则 x=-3t/2带入得到
lim (1+f(x)/x)^(1/x) (1+(-2x/3)) ^(1/x) (1+t)^(1/(-3t/2)) = [(1+t)^(1/t)] ^(-2/3) = e^(-2/3)