Thursday, 14 September 2017

calculus - Showing that limlimitsx0fracxsin(1/x)cos(1/x)x does not exist without sequence

Show that lim does not exist.




I understand that at 0, the \dfrac{\cos(1/x)}x term varies between (-\infty , + \infty). But I want a complete formal proof that use the definition of limit (ε, δ) and not using sequences like (2k\pi n) or other techniques like l'Hopital, etc… How to write a formal proof for that?



Also I want to show that \lim\limits_{x\to0} (x\sin(1/x) - \cos(1/x)) does not exist without using any sequences and only using the definition of limit.

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real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

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