Two Definitions of Limits, with ExamplesDate: 05/11/98 at 05:23:25 From: Darryl Cain Subject: Limit Theory Dear Dr. Math, I have seen many great articles from your homepage, and was wondering if you could help me with a definition. I need to know what Limit Theory is, and I can't find it anywhere. I know it would be great to hear from you. Thanks a lot, Darryl Cain Date: 05/11/98 at 15:09:27 From: Doctor Rob Subject: Re: Limit Theory I am not aware of a subject with the title "Limit Theory." One does study limits when one takes Calculus. There are several kinds of limits that are studied. One definition is like this: Given a function f(x) from the real numbers to the real numbers, and two real numbers a and L, we say that the limit of f(x) as x approaches a is equal to L, and write lim f(x) = L x->a if, for every epsilon > 0, no matter how small, there exists a delta > 0 such that |x - a| < delta implies that |f(x) - L| < epsilon. You can see that this has to do with values of the function f(x) being near L for all values of x near enough to a. Example: lim sin(x)/x = 1 x->0 because for every epsilon > 0, no matter how small, we can put delta = sqrt(6*epsilon). Then, since x > sin(x) > x - x^3/3! for x > 0, and since x - x^3/3! > sin(x) > x for x < 0, and sin(x) = x for x = 0, we have that: |sin(x) - x| <= |x^3/3!| = (|x|^3)/6 Then whenever |x| = |x - 0| < delta, |sin(x)/x - 1| <= (|x|^2)/6 < (delta^2)/6 <= epsilon Another definition is like this: Given an infinite sequence of real numbers: {x(n): n = 1, 2, 3, ...}, and a real number L, we say that the limit of x(n) as n approaches infinity is equal to L, and write: lim x(n) = L n->infinity if, for every epsilon > 0, no matter how small, there is a natural number N > 0 such that n > N implies that |x(n) - L| < epsilon. You can see that this has to do with the value of terms of the sequence being near L for all large enough values of n. Example: lim (n + 1)/(2*n + 1) = 1/2 n->infinity because, for every epsilon > 0, no matter how small, we can put N > 1/(4*epsilon). Then whenever n > N > 1/(4*epsilon), epsilon > 1/(4*n) > 1/(4*n + 2) = |(n + 1)/(2*n + 1) - 1/2| = |x(n) - L| -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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