A statement equivalent to the definition of limits at infinity?

Question

I was fiddling around with the definition of limits at infinity and believe I have found a statement that is equivalent to the definition.

So the question is this: are the following two statements equivalent?

(1) $\lim_{x\rightarrow\infty}f\left(x\right)=L$

(2) $\exists c>0\exists M>0\left(\sup\left\{ \left|x\left(f\left(x\right)-L\right)\right|:x\geq c\right\} \leq M\right)$

Answer 1

No, they are not equivalent. For a counterexample, take

$$f(x) = \frac {1} {\sqrt x}$$

Your second definition assumes something about the rate of decay of $f$ at infinity; the usual definition assumes nothing.