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I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit but I can't find a closing proof. Anyone that

If $\lim_{x \rightarrow a} | f(x) | = | A |$ then $ \lim_{x \rightarrow a}f(x) = A $

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    $\begingroup$ You should try some negative numbers as well... $\endgroup$ – Berci Jan 24 '13 at 16:00
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    $\begingroup$ $\lim\limits_{x\rightarrow a} f(x)$ need not even exist. For example, take $f(x)=\cases{1,&$x$ rational\cr -1, &$x$ irrational }.$ $\endgroup$ – David Mitra Jan 24 '13 at 16:08
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The problem is that it isn't true. As a concrete example, $f(x) = x.$ Then $\lim_{x\to a} |f(x)| = a = |-a|$ if $a > 0$ but $\lim_{x\to a} f(x) \not = -a.$ The issue pops up in that $|A|$ can be either $A$ or $-A$ depending on the sign of $A.$

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  • $\begingroup$ How could I not see that! Thanks! $\endgroup$ – tim_a Jan 24 '13 at 16:12
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Let $f$ be constant $1$ and $A:=-1$.

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