Subject: Re: Cauchy's limit
Date: Mon, 3 Nov 1997 08:05:43 -0500
From: Alex Bogomolny
I think it took me longer to write the response than it would for you to look into a book. It's a wasting of resources.
(1) |Gx - g| is small as x approaches c
(2) |Fx - f| is small as x approaches c
(3) of course you assume f<>0
From (2,3), for x close enough to c
(4) |Fx| > f/2
(5) |Gx/Fx - g/f| = |(fGx - gFx)/fFx|
(6) fGx - gFx = f(Gx -g) - g(Fx - f)
(7) |fGx - gFx| <= |f||Gx -g| + |g||Fx - f|
Obviously, (7) becomes small as x goes to c.
|Gx/Fx - g/f| <= 2(|f||Gx -g| + |g||Fx - f|)/f^f
which becomes increasingly small.