Suppose that $L$ is a real number and $f$ is a real-valued function defined on some interval $(b, \infty)$. We say that $\displaystyle{\lim_{x \to \infty} f(x) =L}$ if for every positive real number $\epsilon$, there is a real number $M$ such that if $x>M$ then $|f(x) -L| < \epsilon$.
Is this statement correct, or should it be amended to imply that a limit can exist at L (i.e. it is possible for a limit to exist at L), but does not have to be the limit of the function? For example, we can prove from this definition that $\displaystyle{\lim_{x \to \infty} \frac{4}{x^2}=0}$, but can't one also prove that $\displaystyle \lim_{x \to \infty} \frac{4}{x^2}=-0.001$, $\displaystyle \lim_{x \to \infty} \frac{4}{x^2}=-0.0001$ and other false claims by application of this definition?