Expert: Paul Klarreich - 1/20/2006

Question
thanks again for your thoughts.  Im still a little baffled as to which function actually holds WRONG for the wrong definition but yet holds true for the correct definition?   

The GIF function doesnt work.  What function would you give as an example to answer the question?  thanks again

I think we're getting there.  

Answer
Hi, Jerry,

You wrote:
thanks again for your thoughts. Im still a little baffled as to which function actually holds WRONG for the wrong definition but yet holds true for the correct definition?

The GIF function doesnt work. What function would you give as an example to answer the question? thanks again

I think we're getting there.  
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I think I said something along these lines:

There can be NO function that satisfies the 'bad' definition (I will start calling it that) but satisfies the 'good' definition.  

The bad definition says that the limit exists and is equal to L if we can find AT LEAST ONE point x where f(x) is near L.

The good definition says that the limit exists and is equal to L if f(x) is near L for EVERY point near x0.

Obviously if f(x) is near L for every x near x0 then you can find at least one x where f(x) is near L.

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I really do think that the 'bad' definition was supposed to say 'you can find x near x0 where f(x) is near L' or words to that effect.  Perhaps there was a misprint or you left it out, or whatever.

No matter -- the same thinking applies.  Being able to find ONE point where f(x) is near L does not mean they all satisfy f(x) is near L, but the reverse is certainly true.