The limit of sin(1/x) as x approaches 0 does not exist (Proof) [ILIEKMATHPHYSICS]

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This video references "Intro to Real Analysis" by Bartle and Sherbert Fourth Edition. For more details related to the context of this video, see section 4.1. In the book, they prove this limit using the divergence criterion for limits. Here, we do so straight from the definition itself.

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ILIEKMATHPHYSICS

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We can also create two sequences (a_n and b_n), which tends to 0, but f(a_n) and f(b_n) not going to the same limit, thus proving the limit doesn't exist. One way of taking such sequences can just be to take the the numbers where f(x) is =1, and the other case where its -1, which we can solve for.

ansumanc

Can you do this proof using the negation of the definition of a limit?

davidlawler

Mind blown
I need to get this book asap

saaah

These proofs always do my head in. I was thinking what is it in this proof that means you can't apply the same logic to sin(x) limit doesn't exist by saying epsilon = 2 and setting x = k.pi or even setting epsilon = 1? By eliminating the "one over" everywhere gets the same....or is it the archimedian step that is actually key. Sometimes these videos need (possibly) stupid examples to show the value of these proofs.

edcoad

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