原式=x→∞lim{(1/e)-[x/(x+1)]^x}/(1/x)(0/0型)
=x→∞lim{1/(x+1)+ln[x/(x+1)]}/(1/x²)(0/0型)
=x→∞lim{-1/(x+1)²+1/x(x+1)]}/(-3/x²)
=x→∞lim(1/3)[x²/(x+1)²-x/(x+1)]
=x→∞lim(1/3){1/[1+(2/x)+(1/x²)]-1/[1+(1/x)]}
=0
【其中[x/(x+1)]^x的导数的求法如下：设y=[x/(1+x)]^x,两边取对数得lny=x[lnx-ln(1+x)],
两边对x取导数得y'/y=[lnx-ln(1+x)]+x[(1/x)-1/(1+x)]=ln[x/(1+x)]+1/(1+x)=1/(1+x)+ln[x/(x+1)]；】
【其中ln[x/(x+1)]的导数的求法如下：{ln[x/(x+1)]}'=[1/(x+1)²]/[x/(x+1)]=1/[x(x+1)].】