```Date: Jan 26, 2013 5:56 PM
Author: Charles Hottel
Subject: Limit Problem

I am having a problem following an example in my book.I understand the concept of limit but sometimes I get confusedmanipulating expressions with absolute values in them.  Here is the problem:Prove  lim(x->c) 1/x  = 1/c, c not equal zeroSo 0 < | x-c| < delta,  implies |1/x - 1/c| < epsilon|1/x - 1/c| = | (c-x) / {xc}| = 1/|x| * 1/|c| * (x-c) < epsilonFactor 1/|x| is troublesome if x is near zero, so we bound it to keep it away from zero.So |c| = |c - x + x| <= |c-x| + |x| and this imples |x| >= |c| - |x-c|I think I understand everything up to this point, but  not the next steps, which areIf we choose delta <= |c|/2 we succeed in making |x| >= |c| / 2.Finally if we require delta <= [(epsilon) * (c**2)} / 2 then[1/|x| * 1/|c| *  |x-c|]  <  [1 / (|c|/2)]  *  [1/|c|]  *  [((epsilon) * (c**2)) / 2] = epsilonHow did they know to choose delta <= |c|/2?How does that lead to |x| > |c|/2 implies 1/|x| < 1/(|c|/2) ?I did not sleep well last night and I feel I must be missing somethingthat would be obvious if my head was clearer.  Thanks for any help.
```