Denote by \(\lambda\) one-dimensional Lebesgue measure. Prove that

(i) \(\int_{(1, \infty)} e^{-x} \ln (x) \lambda(d x)=\lim _{k ightarrow \infty} \int_{(1, k)}\left(1-\frac{x}{k}ight)^{k} \ln (x) \lambda(d x)\),

(ii) \(\int_{(0,1)} e^{-x} \ln (x) \lambda(d x)=\lim _{k ightarrow \infty} \int_{(0,1)}\left(1-\frac{x}{k}ight)^{k} \ln (x) \lambda(d x)\).