For any function \(f\) and numbers \(a\) and \(L\), we write "as \(x \to a^{\neq}, f(x) \to L\)" to mean that for every number \(\epsilon > 0\), there is some corresponding number \(\delta > 0\) such that if \(0 < |x - a| < \delta\) then \(|f(x) - L| < \epsilon\)

Notes:

• \(|x-a| < \delta\) is equivalent to \(-\delta < x - a < \delta\) or \(a - \delta < x < \delta + a\)
• \(0 < |x-a| < \delta\) means that \(x\) is between \(a-\delta\) and \(a + \delta\), but \(x \neq a\)

There cannot be more than one number \(L\) such that as "x → a^≠, f(x) → L".

If there is a number \(L\) such that as \(x \to a^{\neq}, f(x) \to L\), then we defined \(\lim_{x \to a} f(x)\) to be the unique such number \(L\). If there is no such number \(L\), then \(\lim_{x \to a} f(x)\) is undefined.