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#### sweatingbear

##### Member

- May 3, 2013

- 91

There are just two things left I need to wrap my mind around, after that I think I will have comprehended the epsilon-delta concept.

In example 3 in the document

*epsilon-delta1.pdf*where the task is to show that \(\displaystyle \lim_{x \to 5} \, (x^2) = 25\), they assume that there exists an \(\displaystyle M\) such that \(\displaystyle |x + 5| \leqslant M\).

(1) Is it not supposed to be a strict inequality i.e. \(\displaystyle |x+5| < M\) and not \(\displaystyle |x+5| \leqslant M\)? Why would the eventual equality between \(\displaystyle M\) and \(\displaystyle |x+5|\) ever be interesting?

They make the aforementioned requirement when one arrives at

\(\displaystyle |x-5| < \frac {\epsilon}{|x+5|} \, .\)

We somehow, normally through algebraic manipulations, wish to arrive at \(\displaystyle |x-5| < \frac{\epsilon}{M}\) and in their procedure, they write

\(\displaystyle |x-5||x+5| < \epsilon \iff |x-5|M < \epsilon \, .\)

(2) The steps above have overlooked something. Sure, I can buy that \(\displaystyle |x-5||x+5| < |x-5|M\) because we stipulated an upper bound for \(\displaystyle |x+5|\) but just because \(\displaystyle |x-5|M\) is greater than \(\displaystyle |x-5||x+5|\) does not mean that it

*also*must be less than epsilon, right?

Drawing a number line, one can readily conclude that having a < c and a < b does not imply b < c.

What is going on?