$\displaystyle \lim_{x \to 0}$ cos(pi/x)/(x - 2)

How could I find this limit? Any idea would be appreciable.

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- Jul 5th 2011, 06:05 AMinitMFind limit
$\displaystyle \lim_{x \to 0}$ cos(pi/x)/(x - 2)

How could I find this limit? Any idea would be appreciable. - Jul 5th 2011, 06:10 AMProve ItRe: Find limit
The limit does not exist, because for values infinitessimally close to $\displaystyle \displaystyle x = 0$, the function $\displaystyle \displaystyle \cos{\left(\frac{\pi}{x}\right)}$ can equal any value in $\displaystyle \displaystyle [-1, 1]$ (you can not determine a single value that it goes to).

- Jul 5th 2011, 06:25 AMAlso sprach ZarathustraRe: Find limit
- Jul 5th 2011, 06:34 AMinitMRe: Find limit
- Jul 5th 2011, 08:03 AMHallsofIvyRe: Find limit
Actually, I think there is a slight error there- the "$\displaystyle \pi$" should not be in there. Try instead $\displaystyle x_n= \frac{1}{2n}$ and $\displaystyle y_n= \frac{1}{2n+1}$, 1 over the even integers and 1 over the odd integers.

Now, I am hoping that "It is a little bit difficult for me" does NOT mean "I really don't want to actually do anything my self" so I will make some suggestions: If $\displaystyle x_n= \frac{1}{2n}$, what is $\displaystyle \frac{\pi}{x_n}$? What is the cosine of that? What does that go to as n goes to infinity? If $\displaystyle y_n= \frac{1}{2n+1}$, what is $\displaystyle \frac{\pi}{y_n}$? What is the cosine of that? What does that go to as n goes to infinity?