当n趋近于正无穷时,求lim｛n[ln(n+1)-lnn]｝

Question

当n趋近于正无穷时,求lim｛n[ln(n+1)-lnn]｝

Answer 1

lim(n->∞){n[ln(n+1)-lnn]}
=lim(n->∞){n*ln[(n+1)/n]}
=lim(n->∞){ln[(1+1/n)^n]}
=ln{lim(n->∞)[(1+1/n)^n]}
=lne (应用重要极限lim(z->0)[(1+z)^(1/z)]=e)
=1.