Expert: Paul Klarreich - 1/25/2012

Question
Hi Sir,

 May I know how to prove, using epsilon-delta definition, that limit of sin(1/x) does not exist as x tends to 0?


Regards
Alfredo

Answer
Questioner:Alfredo
Country:Singapore

Hi Sir,

 May I know how to prove, using epsilon-delta definition, that limit of sin(1/x) does not exist as x tends to 0?
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If lim [x->0] f(x) = a, then given e, you can find d such that WHENEVER x is near 0, f(x) is near a.

To FALSIFY this, show that for ANY delta, you can find an x in  [-d,d] where f(x) is not near a, no matter what a you pick.

What e do you pick?  Use the fact that as x -> 0, sin(1/x) will oscillate between +1 and -1.  So if you claim lim [x->0] f(x) = some positive a, take e = 1/2.  In any [-d,d] interval, some f(x) = -1, and abs( f(x) - a ) > 1/2.

You can finish up now.