Useful definition of limit?

Question

Is the following definition of a limit a useful one? Does it make sense? Why/Why not?

$lim_{x\rightarrow a} f(x) = b \Leftrightarrow \forall \varepsilon > 0 : 0<|x-a|<\varepsilon \Rightarrow|f(x)-b|<\varepsilon$

EDIT: Am I wrong in saying that this definition works for functions that converge to their limit faster than x does?

Answer 1

Consider $f(x)=x^3$ and $a=0.$ Take any $\epsilon>0$ you like, and consider what happens when $|x|$ is only slightly less than $\epsilon.$