Help Prove Lim x-infinity cos(nx) = dne

[bluemax43]

Help! Prove Lim x--infinity cos(nx) = dne

Hey guys! I am new here however, I have been lurking around for a while. I need some help with a problem that I am currently working on. Here it is:

Homework Statement

Prove Lim x--infinity cos(nx) = does not exist

The Attempt at a Solution

As of now, I am not quite sure how to approach the problem. I know that I can say that if x is a multiple of 2π then it will converge to 1 however, if it is not a multiple of 2π then it will simply oscillate until infinity. As such, the limit does not exist. However, I doubt that that is what is required of the problem. I am thinking that there must be some systematic setup that I can use to prove this problem otherwise. Is there such a way or is my reasoning correct?

In our other problems, we used epsilon-delta proofs to prove that certain functions converged to p.

Anythings would help! Thanks guys!

 

[frenchkiki]

This reminds of a problem I had to do recently. Here are my thoughts:
Pick 2 sequences (x$_{n}$)$_{n}$ and (y$_{n}$)$_{n}$ such that lim x$_{n}$ = infinity as n tends to infinity and lim y$_{n}$ = infinity as n tends to infinity. Then show that lim f(x$_{n}$) ≠ lim f(y$_{n}$), therefore the limit does not exist.

 

[bluemax43]

This reminds of a problem I had to do recently. Here are my thoughts:
Pick 2 sequences (x$_{n}$)$_{n}$ and (y$_{n}$)$_{n}$ such that lim x$_{n}$ = infinity as n tends to infinity and lim y$_{n}$ = infinity as n tends to infinity. Then show that lim f(x$_{n}$) ≠ lim f(y$_{n}$), therefore the limit does not exist.

So if I understand your statement correctly, I should choose for example cos(2πn) and show that that the limit goes to 1 and then pick any other to show that it goes to some other number? So basically my reasoning was right?

 

[frenchkiki]

So if I understand your statement correctly, I should choose for example cos(2πn) and show that that the limit goes to 1 and then pick any other to show that it goes to some other number? So basically my reasoning was right?

Yes it is correct.

 

[bluemax43]

got it thanks!