Find limit

**initM** · July 5th 2011, 06:05 AM

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

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

**Prove It** · July 5th 2011, 06:10 AM

Originally Posted by initM

$\lim_{x \to 0}$ 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 $\displaystyle x = 0$ , the function $\displaystyle \cos{\left(\frac{\pi}{x}\right)}$ can equal any value in $\displaystyle [-1, 1]$ (you can not determine a single value that it goes to).

**Also sprach Zarathustra** · July 5th 2011, 06:25 AM

Originally Posted by initM

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

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

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

Could you continue?

**initM** · July 5th 2011, 06:34 AM

Originally Posted by Also sprach Zarathustra

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

Could you continue?

It is little bit difficult for me. Can I use any substitution here?

**HallsofIvy** · July 5th 2011, 08:03 AM

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 " $\pi$ " should not be in there. Try instead $x_n= \frac{1}{2n}$ and $y_n= \frac{1}{2n+1}$ , 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 $x_n= \frac{1}{2n}$ , what is $\frac{\pi}{x_n}$ ? What is the cosine of that? What does that go to as n goes to infinity? If $y_n= \frac{1}{2n+1}$ , what is $\frac{\pi}{y_n}$ ? What is the cosine of that? What does that go to as n goes to infinity?

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