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作者  skeleton (數殺之氣 神鬼皆泣) 站內  97RA_I
標題  CH.2 Ex6.
時間  2008/12/01 Mon 03:17:41

(a)             { n^4*x-n^5+n  if x屬於(n-1/(n^3),n] n≧2
     let f(x) = {-n^4*x+n^5+n  if x屬於(n,n+1/(n^3)] n≧2
                {     0           otherwise

  ∵ f(x) = x for x≧2,x屬於N ∴ that is clear limsup f(x) = +∞

        ∞   n+1/(n^3)             ∞              ∞
  ∫f = Σ  ∫         f(x) dx  =  Σ n*(1/n^3) =  Σ 1/n^2 < ∞
        n=2  n-1/(n^3)             n=2             n=2

    f is integrable.  #


(b)  不失一般性 we can assume f≧0 ,

 ˙suppose limsup f(x) ≧ c   for some constant c
            x->∞
  => there exists sequence {x_n} with x_(n+1)-x_n≧1,

       such that lim f(x_n) ≧ c
                 n->∞
  => there exists N>0, s.t. if n≧N, then f(x_n) ≧ (2/3)*c

   since f is uniform continuous, there exists 0 < δ < 1/2,

       s.t. if |x-x_n|<δ, then |f(x)-f(x_n)| < c/3

  => f(x) ≧ |f(x_n)| - |f(x)-f(x_n)| > c/3   if |x-x_n|<δ & n≧N

         ∞   x_n+δ           ∞                       ∞
   ∫f = Σ  ∫     f(x) dx ≧ Σ (c/3)*2δ = (c/3)*2δ*Σ 1 = ∞
         n=N  x_n-δ           n=N                      n=N

  => f isn't integrable   -> <-

  Hence limsup f(x) = 0
         x->∞

  since f≧0,

  => 0 = limsup f(x) ≧ liminf f(x) ≧ 0
          x->∞          x->∞

  => lim f(x) = 0   --------------------------------- (i)
     x->∞

 ˙suppose limsup f(x) ≧ d   for some constant d
           x->-∞
  => there exists sequence {x_n} with x_n-x_(n+1)≧1,

       such that lim f(x_n) ≧ d
                 n->∞
  => there exists M>0, s.t. if n≧M, then f(x_n) ≧ (2/3)*d

   since f is uniform continuous, there exists 0 < δ < 1/2,

       s.t. if |x-x_n|<δ, then |f(x)-f(x_n)| < d/3

  => f(x) ≧ |f(x_n)| - |f(x)-f(x_n)| > d/3   if |x-x_n|<δ & n≧M

         ∞   x_n+δ           ∞                       ∞
   ∫f = Σ  ∫     f(x) dx ≧ Σ (d/3)*2δ = (d/3)*2δ*Σ 1 = ∞
         n=M  x_n-δ           n=M                      n=M

  => f isn't integrable   -> <-

  Hence limsup f(x) = 0
        x->-∞

  since f≧0,

  => 0 = limsup f(x) ≧ liminf f(x) ≧ 0
         x->-∞         x->-∞

  => lim f(x) = 0   --------------------------------- (i)
     x->-∞


  by (i) & (ii), we know  lim  f(x) = lim f(x) = lim f(x) = 0
                         |x|->∞     x->∞      x->-∞

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□ Modify: 2008/12/03 Wed 01:37:42  113-223.dorm.ncu.edu.tw 修改

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