Question

求极限limx→e (lnx-1)/x-e.

答案是1/e. 洛比达没学过><！

Answer 1

lim(x->e) (lnx-1)/(x-e) (0/0)
= lim(x->e) (1/x)/1
=1/e
or
expands lnx about e
lnx = lne +(x-e)/e + (x-e)^2/e^2+...
= 1+(x-e)/e + (x-e)^2/e^2+...
(lnx-1)/(x-e)
= [ 1+(x-e)/e + (x-e)^2/e^2+... - 1] /(x-e)
= ((x-e)/e + (x-e)^2/e^2+.)/(x-e)
= 1/e + (x-e)/e^2 + (x-e)^2/e^3 +...
lim(x->e)(lnx-1)/(x-e)
=lim(x->e)[1/e + (x-e)/e^2 + (x-e)^2/e^3 +...]
=1/e

Answer 2

令t=x-e
原式=lim(t->0) [ln(t+e)-1]/t
=lim(t->0) [lne+ln(t/e+1)-1]/t
=lim(t->0) [ln(t/e+1)]/t
=lim(t->0) (t/e)/t 等价无穷小代换
=1/e