Saturday, 29 July 2017

calculus - Showing a limit does not exist using Cauchy's epsilon,delta limit definition





Show lim does not exist using Cauchy definition of limit.




Am supposed to negate the defintion: \exists \epsilon >0 :\forall \delta>0 :(\forall x: 0<|x-a|>\delta \Rightarrow |f(x)-L| > \epsilon) ?



But here a is infinity and there is no L so I don't know how to write the inequality.


Answer



To show that the limit doesn't exist using the formal definition of a limit, I would do something like this:



Suppose \lim\limits_{x\to\infty}x\sin x=L. Then we should have:

\forall\epsilon>0\, \exists N\in\mathbb{N}:x>N\Rightarrow|x\sin x-L|<\epsilon
However \begin{equation}|x\sin x-L|\geq|x\sin x|-|L|=|x||\sin x|-|L|\end{equation}
Now by choosing x=\frac\pi2+2Nk\pi (k\in\mathbb{Z}_{\geq0}) we can make this quantitiy arbitrairilly high, while still having x>N. This is a contradiction. Hence the limit doesn't exist.


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