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Author:      jay a browne
Could you please explain the definition of limits that use the greek epsilon
and delta.

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 following way:

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.


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