1. ## Limits

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

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

Thanks!

2. Originally Posted by shadow_2145
Use the Divergence Criterion for Functional Limits corollary:
(Let $\displaystyle f$ be a function defined on $\displaystyle A$, and let $\displaystyle c$ be a limit point of $\displaystyle A$. If there exist two sequences $\displaystyle (x_n)$ and $\displaystyle (y_n)$ in $\displaystyle A$ with $\displaystyle x_n \not= c$ and $\displaystyle y_n \not= c$ and lim$\displaystyle x_n$=lim$\displaystyle y_n$=$\displaystyle c$ but lim$\displaystyle f(x_n) \not=$lim$\displaystyle f(y_n)$, then we conclude that the functional limit $\displaystyle lim_{x\rightarrow c} f(x)$ does not exist.)the
to show that the following limit does not exist.

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

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