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 limxf(x) and limxf(x) exist. Show that limxf(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 but that didn't really work out. Now I'm thinking I should assume limxf(x)=L0 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 limxf(x)=L0, the contradiction would come from the mean value theorem (consider f(x)f(M) for a fixed but arbitrary large M, and let x).


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.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection f:AB and I want to get bijection. Can I just resting codomain to f(A)? I know that every function i...