lim x趋于0 (x(e^x+1)-2(e^x-1))/x^3

问题描述:

lim x趋于0 (x(e^x+1)-2(e^x-1))/x^3

lim (x→0) (x(e^x+1)-2(e^x-1))/x^3 (0/0)
=lim (x→0) (e^x+xe^x+1-2e^x)/(3x^2)
=lim (x→0) (xe^x+1-e^x)/(3x^2) (0/0)
=lim (x→0) (e^x+xe^x-e^x)/(6x)
=1/6

用罗比达法则
原式=lim(x->0) (e^x+1+xe^x-2e^x)/3x^2
=lim(x->0) (1+xe^x-e^x)/3x^2
=lim(x->0) (e^x+xe^x-e^x)/6x
=lim(x->0) e^x/6
=1/6