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## Homework Statement

I need to know what's wrong with the following proof:

Assume that [PLAIN]http://img16.imageshack.us/img16/4839/eq1.gif [Broken] [Broken][/URL] exists. In other words:

[PLAIN]http://img8.imageshack.us/img8/1856/eq2.gif [Broken] (1)

But:

[PLAIN]http://img801.imageshack.us/img801/8374/eq3.gif [Broken] (2)

And because sin(1/x) is an odd function:

[PLAIN]http://img24.imageshack.us/img24/8453/eq4.gif [Broken] (3)

Therefore, by (1), if [PLAIN]http://img16.imageshack.us/img16/4839/eq1.gif [Broken] [Broken][/URL] exists, then:

[PLAIN]http://img191.imageshack.us/img191/2339/eq5.gif [Broken]

[PLAIN]http://img215.imageshack.us/img215/3218/eq6.gif [Broken]

Similarly,

[PLAIN]http://img641.imageshack.us/img641/3781/eq7.gif [Broken]

[PLAIN]http://img696.imageshack.us/img696/7108/eq8.gif [Broken]

If [PLAIN]http://img16.imageshack.us/img16/4839/eq1.gif [Broken] [Broken][/URL] exists, the only value at which the limit can exist is 0. Since the limit converges to a single value, the limit exists and is equal to 0.

## Homework Equations

[PLAIN]http://img812.imageshack.us/img812/6119/eq10.gif [Broken]

## The Attempt at a Solution

The Laurent series disagrees. >:(

I know there's something wrong with the proof, since it's well accepted that the limit doesn't exist. I'm just not sure what. Any help is appreciated.

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