Explanation of limits
Author: jay a browne
Could you please explain the definition of limits that use the greek epsilon
First of all you need the notion of a function - the function takes a real
number and returns another real number. The limit is then defined in the
limit f(x) = limiting value of f(x + epsilon) as epsilon --> 0
(that should be value of f(x0 + epsilon)).
This is best illustrated by an
example: say I give you the function:
f(x) = (1 - x^2)/(1 - x)
then for x0 = 0, say, the limit is trivial: f(0) = 1 can be computed just
by substituting in x = 0. But, for x0 = 1, you cannot substitute x = 1 into
f(x) as written - the result would be 0/0 which is indeterminate. However,
if you graph the function f(x) you will notice that it is perfectly well
defined in the neighborhood of x = 1, so we just say the limiting value is
that determined by points arbitrarily close to, but not equal to, 1 -
substitute in x = 1 + epsilon, reduce, and then try and figure out if you
can substitute epsilon = 0 in the result.
What does it mean when we say lim f(x) = L? Roughly speaking, we mean
that for values of x "close to" A, the function f gives values "close to" L.
(For this question, the value of f(A), if in fact f is defined for x = A, is
not relevant.) But in math we have to be more precise than "close to". We
do that by using this so-called delta-epsilon formalism. For brevity, let
us write D for delta and E for epsilon. We say that the above limit means:
given any positive number E, we can find a positive number D such that, if x
is any number satisfying |x-A| < D (and for which f is defined) we have
|f(x)-L| < E. That is, given E we can find a D so that if x satisfies
A - D < x < A + D, then we are guaranteed that L - E < f(x) < L + E. In the
back of our minds we are thinking of E and D as very small positive numbers.
We can consider this graphically too. Plot the function f in the vicinity
of the point (A, L); draw horizontal lines a distance E above and below this
point. The idea is to find a distance D so that, if you draw vertical lines
at x = A - D and x = A + D, the box determined by the four lines [which
contains the point (A, L)] has the property that the graph of f never goes
through the top or bottom of the box. Obviously this distance is not
unique; if you find a value that works, any smaller distance will work too.
Click here to return to the Mathematics Archives
Update: June 2012