A statement equivalent to the definition of limits at infinity?

[phoenixthoth]

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

 

[micromass]

Consider $f(x)=\frac{1}{\log(x)}$ and $L=0$.

 

[Bacle2]

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

Just to nitpick a bit: you're thinking of the limit _as x tends to infinity_, and not quite the limit _at infinity_ , since infinity is not a real number ( at least not a standard real ).