设lim[(x-k)/x)]^(-2x)=limxsin2/x,(x→∞).求k.

问题描述:

设lim[(x-k)/x)]^(-2x)=limxsin2/x,(x→∞).求k.

lim[(x-k)/x)]^(-2x)=lim[1+k/(x-k)]^(2x)=lim[1+1/(x/k-1)]^(2x)=lim[1+1/(x/k-1)]^[2k(x/k-1)+2k]=lim[[1+1/(x/k-1)]^(x/k-1)]^2k*[1+1/(x/k-1)]^2k=e^2klimxsin2/x=lim2sin(2/x)/(2/x)=2即e^2k=2 => k=(ln...