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Math Help - another limit function question..

  1. #1
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    another limit function question..

    f_n(x)= 1 ,1<=x<n
    f_n(x)= 0 ,x>n

    why why whyyyy the border function is 1

    i cant see the reason
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  2. #2
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    Quote Originally Posted by transgalactic View Post
    f_n(x)= 1 ,1<=x<n
    f_n(x)= 0 ,x>n

    why why whyyyy the border function is 1

    i cant see the reason
    You mean the limit function, as in your title. Also your definition isn't complete. Do you mean f_n(x)= 1 for x\le n? Or possibly, f_n(x)= 1 for x< 1 and f_n(x)= 0 for x\ge n? Both give the same limit but without an "=" in one of them some of the functions are not defined for some x. I'm going to assume it is x\le 1.

    This is "point-wise" convergence. We look at an individual value of x, say x= x_0 and the convergence of the numerical sequence {f_n(x_0)}. For example, if x= 2, then f_1(2)= 0 because 2> 1. <br />
f_2(2)= 1 because 2= 2. But f_3(2)= 0 because 2< 3 and, for all n> 3, 2< n so f_m(2)= 0. The sequence \{f_n(2)\} is 1, 1, 0, 0, 0, 0, ... which converges to 0.

    In fact, for any x, there exist some positive integer, N, such that x< N (that's the "Archimedian property" of the positive integers). For any such N, f_N(x)= 0 and for all n> N, f_n(x)= 0. The sequence {f_n(x)} consists of some finite number of "1"s followed by an infinite sequence of "0"s and that converges to 0.
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