当x→e，求(lnx-1)/（x-e）
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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
= 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
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用洛必达法则
lim (1/x)/1=1/e
x→e
lim (1/x)/1=1/e
x→e
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