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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!

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# Contradiction of statement regarding monotonicity

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