# Find limit

• Jul 5th 2011, 06:05 AM
initM
Find 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 AM
Prove It
Re: Find limit
Quote:

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

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

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 AM
Also sprach Zarathustra
Re: Find limit
Quote:

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

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

Take $\displaystyle x_n=\frac{1}{2\pi n}$ and $\displaystyle y_n=\frac{1}{(2n+1)\pi}$ and $\displaystyle x_n ,y_n \to 0$ when $\displaystyle n\to \infty$.

Could you continue?
• Jul 5th 2011, 06:34 AM
initM
Re: Find limit
Quote:

Originally Posted by Also sprach Zarathustra
Take $\displaystyle x_n=\frac{1}{2\pi n}$ and $\displaystyle y_n=\frac{1}{(2n+1)\pi}$ and $\displaystyle x_n ,y_n \to 0$ when $\displaystyle n\to \infty$.

Could you continue?

It is little bit difficult for me. Can I use any substitution here?
• Jul 5th 2011, 08:03 AM
HallsofIvy
Re: Find limit
Quote:

Originally Posted by initM
It is little bit difficult for me. Can I use any substitution here?

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?