In the limits and continuity chapter in my textbook, the following definitions are given:
(1) For limit of f(x) as x tends to infinity:
We say that $ƒ(x)$ has the limit $L$ as $x$ approaches infinity and write $\lim_{x\to \infty} f(x) = L$ if, for every number $\epsilon > 0$ there exists a corresponding number $M>0$ such that for all $x>M, |f(x) - L| < \epsilon$.
(2) For infinite limits:
We say that $ƒ(x)$ approaches infinity as $x$ approaches $x_0$, and write $\lim_{x\to x_0 } ƒ(x) = \infty$, if for every number $B>0$ there exists a corresponding $\delta > 0$ such that for all $x$, $$0 < |x - x_0| < \delta \implies f(x) > B$$
What I don't understand is why the numbers M and B have to be greater than zero in the first and second definitions respectively. I think the definition would work without this added constraint, i.e. if M (and, similarly, B) is any real number. So, I don't understand why this condition is present in the definitions.
Since I couldn't find any discussion on this topic anywhere on the net or in my book, I have asked this question.
What am I missing?