Definition of the limit of a function

[Grant Bonner]

Define f: (0,1)-> R by f(x) = x cos (1/x). Does f have a limit at 0?

I have no idea how to do this either. I can see that as x approaches 0, that the fuction oscillates more quickly. I have the proof for sin (1/x) and know that there is no limit as x approaches 0, however the x cos x throws me off. I have a feeling that this fuction does have a limit at 0 and that the limit is 0.

 

[DrLee]

Define f: (0,1)-> R by f(x) = x cos (1/x). Does f have a limit at 0?

I have no idea how to do this either. I can see that as x approaches 0, that the fuction oscillates more quickly. I have the proof for sin (1/x) and know that there is no limit as x approaches 0, however the x cos x throws me off. I have a feeling that this fuction does have a limit at 0 and that the limit is 0.

We know that |cos(1/x)| <= 1 for all x in (0,1). Therefore, as x->0, |x cos(1/x)| <= |x| for all x on (0,1). Therefore f(x)->0 as x->0.