ln [(1+ 1/x)^x]=x* ln (1+ 1/x)=ln (1+ 1/x) /(1/x)
lim(x趋于零)[ln (1+ 1/x) /(1/x)]
=lim(x趋于零){ [1/(1+ 1/x)*(-1 /x^2)] /(-1 /x^2) }
=lim(x趋于零)[1/(1+ 1/x)]=0
所以 lim(x趋于零)[(1+ 1/x)^x]=e^0=1