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
and assume that
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