Does "limit not equal X" mean that the limit exist? If we say that $\lim_{n\to \infty}a_n\not=X$, does it implicitly mean that the limit exists?
In other words, does this negation means:


*

*"it's not true that 'X is the limit' " (and maybe there is not a limit at all)


or


*

*"the limit doesn't equal X" (and therefore, there is a limit, because otherwise, we couldn't speak about it)

 A: Pragmatically, I think most people would tend to understand it as your first statement. You could say that if the limit does not exist, it is in particular not equal to anything. 
You could of course also argue that the expression is meaningless, not well-typed, etc, so that noone "shouldn't" ever write anything like that unless the limit exists. 
Nonetheless, the conclusion of the second of your assertions would probably never be used by anyone who is also saying something useful about the problem. 
(It seems to me that the core of this discussion is the interpretation of the $\not =$ sign. One option is that inequality is a primitive notion that is only applicable to elements of the same set or class, and the other is that it represents the negation of equality, and thus has no such requirements.)
But most importantly, knowing that the limit exists without knowing anything else is probably not that interesting anyway -- and if it would be interesting, you would probably want some better arguments than nitpicking about inequalities!
A: Formally $\lim_{n\to\infty}a_n\neq X$ makes no sense if there is no limit. 
So considering it as a statement that does make sense you can conclude that the limit exists. 
However, if you meet statements like this, be careful anyway. 
A: This is a pet peeve of mine. If I were in charge we'd always write $a_n\to a$ instead of $\lim a_n=a$, because this issue doesn't arise with the negation $a_n\not\to a$.
