Question

用洛必达法则求limx→0(1/x-1/e^x-1)的详细步骤

Answer 1

limx→0(1/x-1/e^x-1)
=limx→0(e^x-1-x)/[x(e^x-1)] (运用等价无穷小代换）
=limx→0(e^x-1-x)/x^2　　（0/0，运用洛必达法则）
=limx→0(e^x-1)/(2x) (运用等价无穷小代换）
=limx→0 x/(2x)
=1/2

Answer 2

e^x-1和x是等价无穷小
limx→0(1/x-1/(e^x-1))
=limx→0((e^x-1)-x)/(e^x-1)x(无穷小)
=limx->0(e^x-1-x)/x^2
洛必达
=limx->0(e^x-1)/2x(无穷小)
=limx->0 x/2x
=1/2

Answer 3

先通分，然后用两次洛必达法则
lim(x→0) [1/x-1/(e^x-1)]
=lim(x→0) [(e^x-1-x)/x(e^x-1)] .........0/0型，用洛必达法则
=lim(x→0) [(e^x-1)/(e^x-1+xe^x)] .........0/0型，用洛必达法则
=lim(x→0) [(e^x)/(2e^x+xe^x)]
=1/2

Answer 4

通分
=lim(e^x-1-x)/x(e^x-1)
=lim(e^x-1)/(e^x-1+x*e^x)
还是0/0
=lim(e^x)/(e^x+e^x+x*e^x)
=lim1/(2+x)
=1/2