Limits

**shadow_2145** · June 5th 2008, 08:12 AM

Use the Divergence Criterion for Functional Limits corollary:
(Let $f$ be a function defined on $A$ , and let $c$ be a limit point of $A$ . If there exist two sequences $(x_n)$ and $(y_n)$ in $A$ with $x_n \not= c$ and $y_n \not= c$ and lim $x_n$ =lim $y_n$ = $c$ but lim $f(x_n) \not=$ lim $f(y_n)$ , then we conclude that the functional limit $lim_{x\rightarrow c} f(x)$ does not exist.)the
to show that the following limit does not exist.

$lim_{x\rightarrow 0}$ $|x|/x$

Thanks!

**Jhevon** · June 5th 2008, 08:29 AM

Originally Posted by shadow_2145

Use the Divergence Criterion for Functional Limits corollary:
(Let $f$ be a function defined on $A$ , and let $c$ be a limit point of $A$ . If there exist two sequences $(x_n)$ and $(y_n)$ in $A$ with $x_n \not= c$ and $y_n \not= c$ and lim $x_n$ =lim $y_n$ = $c$ but lim $f(x_n) \not=$ lim $f(y_n)$ , then we conclude that the functional limit $lim_{x\rightarrow c} f(x)$ does not exist.)the
to show that the following limit does not exist.

$lim_{x\rightarrow 0}$ $|x|/x$

Thanks!

take $f(x) = \frac {|x|}{x}$ , $c = 0$ , $x_n = \frac 1n$ and $y_n = - \frac 1n$

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