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Let I be an open interval that contains the point c and let f be a function that is defined on I except possibly at the point c. Suppose that lim |f(x)| as x->c exists. Give an example to show that lim f(x) as x->c may not exist.

Not really sure of an example for this?

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up vote 4 down vote accepted

$$f(x) = \left\{\begin{array}{lr} -1 & x < 0 \\ 1 & x > 0\end{array}\right.$$

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So what's the limit of |f(x)| here? 0? then the lim f(x) DNE? –  Arnold Oct 21 '13 at 4:54
    
@Arnold What is $|f(x)|$ if $x \ne 0$? –  user61527 Oct 21 '13 at 4:54
    
Wouldn't you want x to be greater than or equal to 0 for one of the parts of the piecewise function? –  Arnold Oct 21 '13 at 4:59
    
@Arnold I just didn't define it at $0$, since there's no need to. –  user61527 Oct 21 '13 at 5:00

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