How do you decide what two paths to take? For example:

Can I pick any paths I want? For example:

The line

And the line

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- Oct 13th 2010, 09:17 PMdownthesun01Quick question about two-path limit test
**How do you decide what two paths to take? For example:**

**Can I pick any paths I want? For example:**

**The line**

**And the line**

- Oct 14th 2010, 12:48 AMProve It
What point are you making this function tend to?

- Oct 14th 2010, 12:50 AMdownthesun01
Sorry. (0,0)

- Oct 14th 2010, 01:00 AMProve It
- Oct 14th 2010, 01:14 AMdownthesun01
Thanks but we haven't learned about polars yet. If someone could explain it for me by using the two-path method, I would much appreciate it. Thanks

- Oct 14th 2010, 04:12 AMHallsofIvy
The original question was "how do you decide which two paths to take". The answer is, you

**don't**. You**try**different paths, perhaps many, until you find two which give different limits. If there**exist**different paths that give different limits, then the limit itself does not exist.

The problem with that is that you can't try**all**paths and there is no way of knowing in advance which, if any, paths will give different limits. Even if you tried**all**straight lines, by using "y= mx" with m left variable, and found that you got the same limit for all straight line paths, that is not enough. There are functions which give the same limit for all**straight**paths but different limits for curved paths and so do NOT have a "limit".

If, for some paths, you get different limits then you know that the limit itself does not exist. If, however, you get the same limit for every path you try, you still do NOT know whether or not the limit itself exists.