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Linneasaur
 2 years ago
Find a continuous f on the interval (0, inf) such that the integral of f(x) on (0, inf) exists, but the limit as x> infinity =/= 0
Linneasaur
 2 years ago
Find a continuous f on the interval (0, inf) such that the integral of f(x) on (0, inf) exists, but the limit as x> infinity =/= 0

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kbomeisl
 2 years ago
Best ResponseYou've already chosen the best response.0cosine, sine, lots. actually any function that can be dissolved vie fourier series into plain cos and sine will fit these parameters. integral will be zero. Did you want a nonzero integral?

kbomeisl
 2 years ago
Best ResponseYou've already chosen the best response.0Maybe you like that answer. Otherwise, if you want an answer with a nonzero integral simply make your function of the form sine(x) + g(x); where g(x) is any function with an existing integral. It doesn't matter whether the limit of g(x) approaches zero when x> infinity, because sine(x) +g(x) will still have a nonconvergent limit (the limit will not exist). If you want the integral of your answer to be X, simply choose a g(x) that has integral X. The integral like the derivative is a linear operator and therefore obeys the first law of linearity ( int[f(x)+g(x)] = int[g(x)]+ int[f(x)]. A less clever way to do this that a lay mathematician may think would work is simply attaching the function g(x) to the function sine(x) (or cos (x)) like making a composite function of the form: from 0 to x inclusive, g(x), and from x to infinity exclusive, sine(x). However, this will not work as at the point that sin(x) is joined to g(x), there will be a cusp where the left limit will not equal the right limit. Right, that is all you will ever have to know about limit theory as a context to this problem. Though it is a fascinating area to specialize in and learn more about. A family of sinusoidal functions is also a perfect place to use a metalimit that approaches sine of cos. But that is a relatively new (very new) concept that hasn't quite caught on yet. Stick to the conventional epsilon delta proof of limits if you want to verify that I am right about all this for your professor. I've done all this in my head, if you want the delta epsilon proof, I can supply this.
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