作者stillboy (joey)
看板trans_math
标题Re: [极限]
时间Wed Oct 8 16:00:25 2008
嵇o※ 引述《Qmmm (Q蛆蛆)》之铭言：
1.lim (X^π - π^X)
x->∞
X^π
=lim π^X ( ------ - 1 )
x->∞ π^X
X^π
=lim π^X ( 0 - 1 ) ( lim ------ = 0_)
x->∞ x->∞ π^X
= -∞ (∵ ∞ * M = ∞ ,M为有界)
所以此极限不存在
: 2. 1 1 1
: lim --------- ( --------- - --- ) = ?
: x->0+ x^(1/3) sin x x
:
x- sinx
= lim x^(-1/3) * (-------------)
x->0+ x sin x
x - sinx
= lim -------------
x->0+ x^(4/3) sinx
x - sinx (1/6) x^3 x 1
= lim -------------* --------- * -------- * --------
x->0+ (1/6) x^3 x sinx x^(4/3)
(1/6)x^3 x x - sinx
= lim ----------- (∵ lim ------- = 1 = lim ----------- )
x->0+ x^(7/3) x->0+ sinx x->0+ (1/6) x^3
= 0
3.
: lim [x^(1/3)]*e^(cosπ/x) = ?
: x->0
e^(-1/x)
= lim ------------
x->0 x^(-1/3)
L'H e^(-1/x) * (1/x^2)
= lim -------------------- = (在L'H一次) = (-3) * ∞ *∞ = -∞
x->0 (-1/3) * x^(-4/3)
所以此极限不存在
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