求极限limx→e (lnx-1)/x-e.答案是1/e. 洛比达没学过><!

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