also, just to point out: there is nothing special about 1/2, we can pick ANY a with 0 < a < 1, in which case:
if we call this number L, what can we say about x for which:
?
(choosing a close to 1 will make L very large, which will make our choice of delta fairly small. that's ok. choosing a close to 0 will make L a bit larger than 1, and whether or not a is smaller than √(ε/L) depends on how small ε is. choosing a = 1/2 represents a "middle ground" where the numbers are easy to calculate with, in this case, and we can focus on ε, rather than the expressions involving a, which can be distracting. but a = 1/3, or a = 1/4, for example, would work just as well).
slight quibble:
since we are finding a LIMIT, rather than checking for continuity, we want a DELETED neighborhood of 0, that is: 0 < |x| < δ. in this example, it doesn't matter, the function being checked:
f(x) = 1/(x^{2} - 1) is continuous (but unbounded) on (-1,1), so on any sub-interval.
but, in general, f(0) doesn't have to exist for the limit of f(x) as x approaches 0 to exist, and even if f(0) does exist, it may not equal the limit. that is: for CONTINUOUS functions, you can find the limit:
by just finding f(0), but if one does not know whether a function f is continuous at 0 or not, to find the limit you don't let x "go to 0", it only gets "near it".
the classic example of this is:
.
note we cannot consider values in any interval containing 0, since x/x is undefined at 0. if we define:
g(x) = x/x , x ≠ 0
g(0) = 5
g is a perfectly good function, and the limit is still 1 at 0, but it isn't true that for ALL x with |x| < δ, |g(x) - 1| < ε, there is one exception: x = 0, in which case
|g(x) - 1| = 4 (and so if ε < 4, we're toast).