Wednesday, May 9, 2018

calculus - Proving that limlimitsxtoinftyf(x)=0 when limlimitsxtoinftyf(x) and limlimitsxtoinftyf(x) exist



I've been trying to solve the following problem:


Suppose that f and f are continuous functions on R, and that lim and \displaystyle\lim_{x\to\infty}f'(x) exist. Show that \displaystyle\lim_{x\to\infty}f'(x) = 0.


I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to 0 as x went to \infty but that didn't really work out. Now I'm thinking I should assume \displaystyle\lim_{x\to\infty}f'(x) = L \neq 0 and try to get a contradiction, but I'm not sure where the contradiction would come from.


Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks


Answer



Hint: If you assume \lim _{x \to \infty } f'(x) = L \ne 0, the contradiction would come from the mean value theorem (consider f(x)-f(M) for a fixed but arbitrary large M, and let x \to \infty).


Explained: If the limit of f(x) exist, there is a horizontal asymptote. Therefore as the function approaches infinity it becomes more linear and thus the derivative approaches zero.


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