The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Topic: Limit Problem
Replies: 3   Last Post: Jan 27, 2013 12:31 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Charles Hottel

Posts: 10
Registered: 9/5/11
Limit Problem
Posted: Jan 26, 2013 5:56 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

I am having a problem following an example in my book.
I understand the concept of limit but sometimes I get confused
manipulating expressions with absolute values in them. Here is the problem:

Prove lim(x->c) 1/x = 1/c, c not equal zero

So 0 < | x-c| < delta, implies |1/x - 1/c| < epsilon

|1/x - 1/c| = | (c-x) / {xc}| = 1/|x| * 1/|c| * (x-c) < epsilon

Factor 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 are

If 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] = epsilon

How 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 something
that would be obvious if my head was clearer. Thanks for any help.

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.