Contradiction of statement regarding monotonicity

[oferon]

Hi all!
We were given to proove or falsify the following statement:

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

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

Now i thought of another one: $$f(x)=\frac{sin(x)+2}{x^2}$$
Is that a good example? Thank you!

 

[LCKurtz]

Hi all!
We were given to proove or falsify the following statement:

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

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

Now i thought of another one: $$f(x)=\frac{sin(x)+2}{x^2}$$
Is that a good example? Thank you!

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.

 

[oferon]

Will do. Thank you!