An alternate definition of Limit Let's say we were to define the concept of limit as follows:
$\displaystyle{\lim_{x \to c}}f(x)=L$ means that for every $x$ in the domain of $f$, there exists an $x_0 \neq x$ in the domain of $f$ such that:
$$|x-c|>|x_0-c|$$and$$|f(x)-L|\ge|f(x_0)-L|$$
I have two questions:

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*Does there exist a limit that can be proven using the (ε, δ)-definition but not by using the above definition?

*Does there exist a limit that can't be proven using the (ε, δ)-definition but can be proven using the above definition?


$L$ can even not be in the range of $f$.
Take for instance $f(x)=1_{\mathbb Q}-1_{\mathbb R\setminus\mathbb Q}$ and $L=0$.
We have $f(x)=\pm 1$ so $f$ cannot converge in traditional sense, but since $|f(x)-L|=\text{cst}$ and you have set for a loose inequality, your condition is always fulfilled.

This is a tortured function but even take $f(x)=\ell$ then by your definition $\lim\limits_{x\to c}f(x)=2\ell$ (or any $u\in\mathbb R$ in fact). That fact that constant functions converge to any limit is kind of "annoying".
