求极限x→0 lim[1/x-1/(e^x-1)]
x→0 lim[1/x-1/(e^x-1)] (∞-∞型)
=x→0 lim(e^x-1-x)/[x(e^x-1)] (0/0型,用罗比塔法则)
=x→0 lim[(e^x-1)/(e^x-1+xe^x) (0/0型,继续用罗比塔法则)
=x→0 lim[e^x/e^x+e^x+xe^x)]=1/2.
求极限x→0 lim[1/x-1/(e^x-1)]
x→0 lim[1/x-1/(e^x-1)] (∞-∞型)
=x→0 lim(e^x-1-x)/[x(e^x-1)] (0/0型,用罗比塔法则)
=x→0 lim[(e^x-1)/(e^x-1+xe^x) (0/0型,继续用罗比塔法则)
=x→0 lim[e^x/e^x+e^x+xe^x)]=1/2.