What does it mean that *a/b* is close to *a/c*
whenever *b* is close to *c*? The real issue
here is that *a/b* depends *continuously* on
*b* as long as *b* is non-zero. Calculus
teachers spend a long time defining and explaining
continuity and some day there may be a lucid and detailed
explanation in these pages. But for the time being let me
just give you what really amounts to a **definition** of
continuity. It makes perfect sense to me, but may seem
unduly arcane to you.

For non-zero *b* and any number *a*, *a/b
* depends continuously on *b* since for any positive number *
epsilon* we can find a positive number *delta*
such that *a/b* differs from *a/c* no more
than *epsilon* as long as *b* does not differ
from *c* by more than *delta.*

In other words we can make the difference in the ratios *
arbitrarily* small by picking the difference in the
denominators *sufficiently* small.

For the remainder of this page let me assume that you know Calculus. Consider a ratio of two functions

*f(x)/g(x)*

and assume that

*f(0)=g(0)=0.*

Then the limit of *f(x)/g(x)* as *x* tends to
zero may exist. Moreover, if both *f* and *g*
are differentiable, and the limit *f'(x)/g'(x)*
exists, then

lim *f(x)/g(x)* = lim *f'(x)/g'(x)*

as *x* tends to zero.

This is known as **the Rule of L'Hôpital.**

Fine print, your comments, more links, Peter Alfeld, PA1UM

[18-Apr-1997]