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Contradiction of statement regarding monotonicity

  1. Feb 12, 2012 #1
    Hi all!
    We were given to proove or falsify the following statement:

    Given [tex]f(x)>0 \,\ ,\,x>0 \,\,\,\,,\lim_{x\to\infty}f(x)=0[/tex]
    Then f(x) is strictly decreasing at certain aεℝ for every x>a

    Now in their solution they contradicted the statement with:
    [tex]\newcommand{\twopartdef}[4]
    {
    \left\{
    \begin{array}{ll}
    #1 & \mbox{if } #2 \\
    #3 & \mbox{if } #4
    \end{array}
    \right. } f(x) = \twopartdef { \frac{1}{2x} } {x \,\,\, rational} {\frac{1}{x}} {x \,\,\, irrational}[/tex]

    Now i thought of another one: [tex] f(x)=\frac{sin(x)+2}{x^2} [/tex]
    Is that a good example? Thank you!
     
  2. jcsd
  3. Feb 12, 2012 #2

    LCKurtz

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    Yes, that's a nice example too. If it is something to hand in you would want to include an argument to show that it isn't strictly decreasing for x large enough.
     
  4. Feb 12, 2012 #3
    Will do. Thank you!
     
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