Question

设f(x)=(e＾x+x)＾1/x,极限x趋于正无穷大[f(x)]＾a=f(1),求常数a

Answer 1

f(1)=(e+1)

limx->+00 [(e^x+x)^(1/x)]^a=e+1

设

y=[(e^x+x)^(1/x)]^a

lny=aln(e^x+x)^(1/x)

设g=ln(e^x+x)^(1/x)

lng=ln(e^x+x)/x

lim(x->+00)ln(e^x+x)/x=lim (e^x+1)/(e^x+x)=lim e^x/(e^x+1)=lim (1/(1+1/e^x))=1

所以 x->+00 时

lng=1

lny=a

y=e^a=e+1

a=ln(e+1)