I need the limit of x to infinity on the problem (cos(2/x))^(x^2) (the quantity cos of 2/x raised to the x squared). My gut told me it would be one, however the true answer is around .1353. How do you solve it?

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- Jan 3rd 2010, 01:09 PMHelpaguyoutLimit as x goes to infinity on hard problem
I need the limit of x to infinity on the problem (cos(2/x))^(x^2) (the quantity cos of 2/x raised to the x squared). My gut told me it would be one, however the true answer is around .1353. How do you solve it?

- Jan 3rd 2010, 03:32 PMAbu-Khalil
But given any number as your function is and on any , your function is . How could that limit exists?

Maybe, you need ... and L'Hôpital does the job. - Jan 3rd 2010, 03:39 PMKrizalid
But abu.

- Jan 3rd 2010, 03:53 PMJose27
I don't believe this question is correct since clearly has negative values in any interval of the form where therefore the usual definition of does not apply (at least in ). Are you by any chance talking about ?

- Jan 3rd 2010, 04:13 PMHelpaguyout
I've checked, and the problem is doable. I believe it is done by constructing a new limit then somehow substituting in, but I don't know how to do that

- Jan 3rd 2010, 04:22 PMJose27
- Jan 3rd 2010, 04:32 PMHelpaguyout
Ugnh, I feel stupid. It's supposed to be 2/x, sorry guys

- Jan 3rd 2010, 05:03 PMKrizalid
Ah, okay, makes sense.

We proceed as follows: as (Very easy to prove, just put ) So

as

Where the important fact here is that as - Jan 4th 2010, 12:02 AMsimplependulum

we know that if , then

therefore

replace with , which also tends to infinity , too .

the limit