Tricky question that deals with big O notation and little o notation

[coder752]

Hi,

I have this question that seems trivial to me.

I need to find a function which is not o(1) but where f(2009) = 0 <---fyi this is a zero
using the fact that f(x) is a function such that f(x) = O(1) <---fyi this is big O notation


I was thinking the following:

f(x) = 2009/x

and make x equal to 0....but it doesn't satisfy for f(2009) part.

My intuition is that I need some sort of function that deals with the 2009 since the 2009 plays a part at the second part which somehow has to get 0 as an answer.

Anyway, thanks so much in advance for help.

Help is greatly appreciated,

coder752

[Zachm]

I need to find a function which is not o(1) but where f(2009) = 0 <---fyi this is a zero
using the fact that f(x) is a function such that f(x) = O(1) <---fyi this is big O notation

Your definition is not clear to me. What is the difference between the function marked in green and the function marked in red ? are they the same function ? different functions ?

Does the function you need to find must make use of the f(x) function ?

Regards,
Zachm

[coder752]

Ok, I'll give you the whole statement.

This is a two parter question to be treated separate.
Suppose f(x) is a function such that f(x) = O(1).

Can e^{f(x)} = o(1)? Yes or no, provide explanation.

Find one such function f(x) which is not o(1) but where f(2009) =0.

Hope that cleared it up for your eyes.

[Zachm]

Suppose f(x) is a function such that f(x) = O(1).

Can e^{f(x)} = o(1)? Yes or no, provide explanation.

You should check the little-O notation definition in order to answer this.

If e^(f(x)) = o(1) then it would mean that:

   lim {x->∞} ( (e^(f(x)) / 1 ) = lim{x->∞} ( e^(f(x) ) = 0

This would be true if and only if:

  lim {x->∞} ( f(x) ) = -∞

But f(x) = O(1) and therefore (big-O definition):

 there exists some constant C and a positive real number r such that for each x > r, 
|f(x)| < C

This would be in contradiction with this demand from f(x):

  lim{x->∞}( f(x) ) = -∞

Thus, such a function doesn't exist.

Find one such function f(x) which is not o(1) but where f(2009) =0.

This is still not clear to me,
- f(x) is not o(1), is it O(1) ?
- Does this requirement still hold ?

e^{f(x)} = o(1)

Regards,
Zachm

[coder752]

I suppose it is.

Does that help any bit?

Basically I believe its asking for a function that is not little o(1) but when you plug in the number 2009 in the new function it will equal to 0 thus agrees both situations.