I keep coming across two different kinds of answers to this question.
The first definition:
We say that $$\lim_{x\to \infty} f(x) = L$$ if the following condition is satisfied:
for every number $\epsilon > 0$ there exists a number $R$, possibly depending on $\epsilon$, such that if $x > R$, then $x$ belongs to the domain of $f$ and $$|f(x) - L| < \epsilon.$$
The second definition is exactly the same as the first one, except there exists a number $R > 0$ rather than "there exists a number $R$".
I understand the first definition, but not the second one because of the restriction on $R$.
Which definition is correct, why would that definition be correct, and why would the other definition be incorrect?
The two are equivalent. If $\exists$ $R \in \Bbb R$ such that for all $x > R$, $|f(x) - L| < \epsilon$, then if $R >0$ there's nothing to prove, and if $R \le 0$, take $R' = 1$, then $R' > 0$ and if $x>R'$, then $x > R$ and so $|f(x) - L| < \epsilon$.