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
Stacey Warren - Expert brainly.com
Hey! We 've verified this expert answer for you, click below to unlock the details :)
At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
cosine, 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 non-zero integral?
Maybe you like that answer. Otherwise, if you want an answer with a non-zero 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 non-convergent 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 meta-limit 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.