I'm kinda stuck on this problem:
Use the definition of functional limit to prove that the limit as $x\to 0$ of $\frac{x}{|x|}$ does not equal $-1$.
Here is the definition of a functional limit:
Let $f\colon A \to \Bbb R$, and let $c$ be a limit point of $A$. Then we say that $\lim_{x\to c} f(x) = L$ if for any $\epsilon >0$, there exists $\delta > 0$ such that if $0 < |x-c| < \delta$, then $|f(x) - L| < \epsilon$.
I've tried several ways, but none of them worked and I'm not sure what to do. Any help would be great! Detailed answers help me with the material a lot! Thanks in advanced, I really appreciate it!
P.S. I apologize in advanced if the wording and layout funny, I'm still learning editing on this forum. Thanks!
Hints: look at the one sided limits and
$$\frac x{|x|}=\begin{cases}1&,\;\;x>0\\{}\\-1&,\;\;x<0\end{cases}$$