The problem statement, all variables and given/known data I know what a limit is and I understand the idea behind it, but I am misunderstanding something in the formal definition of a limit. DEFINITION Let f(x) be defined on an open interval about x0, except possibly at x itself. We say that the limit of f(x) as x approaches x0 is the number L and write limx → x0 f(x) = L if, for every number ε > 0 there exists a corresponding number δ > 0 such that for all x, 0 < abs(x – x0) < δ => abs(f(x) – L) < ε The attempt at a solution For example, take f(x) = x2 I could, incorrectly, assert that limx → 2 f(x2) = 1 Now, δ > abs(x - 2) > 0 and ε > abs(f(x) - 1) If: 1) this holds true for all values of x (except x = x0 = 2), and, 2) for every possible value of epsilon some value of x which satisfies the above statement for both ε and δ exists, Then: 1 is the limit of x2 as x approaches 2. As an example, I pick x = 3. So, δ must be > 3 and ε must be > 8 I can continue and choose arbitrary values of x, none of which seem to be a problem. Likewise, for any value of epsilon I can think of, I can find an value of x which satisfies the inequality ε > abs(f(x) - 1) which also satisfies δ > abs(x - 2) > 0. ex: when ε > .01 I could have x = .999. This x also satisfies δ > abs(x - 2) > 0 (as far I can tell, it's impossible not to satisfy gamma, as long as x isn't equal to x0) By that reasoning, the claim I made above is true and limx → 2 f(x2) = 1 By the reasoning I'm using with the formal definition, I can state that any value of L is the limit for any function at an arbitrary x0 (given that the function has values infinity close to L at some point in time). This is clearly incorrect. Would anyone please point out the error in my application of the definition of a limit? Thank you.