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.
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.
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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.
Mind blown
I need to get this book asap
I have it, its a very good book, along with Principles of Mathematical Analysis by Rudin
beautiful proof!
Can you do this proof using the negation of the definition of a limit?
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.
I love you