令t=1/x则t→0
再令k=lim ｛[1+2^(1/x)+3^(1/x)+…+100^(1/x)]/100｝^(100x)
k=lim[(1^t+2^t+...100^t)/100]^(100/t)
lnk=lim100ln[(1^t+2^t+...100^t)/100]/t
=100lim(ln1*1^t+ln2*2^t+...ln100*100^t)/(1^t+2^t+...+100^t){罗必达法则}
=100*(ln1+ln2+ln3+...ln100)/(1+1+...+1)
=ln(100!)
于是k=100!