Definition: Limit of a Function
Let \(a \in \mathbb{R}\). Let \(I\) be an open interval containing \(a\). Let \(f\) be a real function defined everywhere on \(I\) except possibly at \(a\). Then \(f(x)\) is said to converge to \(L\) as \(x\) approaches \(a\) if and only if for every \(\epsilon > 0\), there is a \(\delta > 0\) (which in general depends on \(\epsilon, f, I\) and \(a\)) such that
$$ 0 < |x - a| < \delta \quad \text{ implies } \quad |f(x) - L| < \epsilon. $$
In this case, write
$$ \lim_{x \to a} f(x) = L. $$
Definition: One Sided Limit
Let \(a \in \mathbb{R}\) and \(f\) be a real function. Then
  1. \(f(x)\) is said to converge to \(L\) as \(x\) approaches \(a\) from the right if and only if \(f\) is defined on some open interval \(I\) with left endpoint \(a\) and for every \(\epsilon > 0\), there is a \(\delta > 0\) (which in general depends on \(\epsilon, f, I\) and \(a\)) such that
    $$ a + \delta \in I \quad \text{ and } \quad a < x < a + \delta \quad \text{ imply } \quad |f(x) - L| < \epsilon. $$
    In this case, we call \(L\) the right-hand limit of \(f\) at \(a\) and denote it by
    $$ \lim_{x \to a+} f(x) = L. $$
  2. \(f(x)\) is said to converge to \(L\) as \(x\) approaches \(a\) from the left if and only if \(f\) is defined on some open interval \(I\) with right endpoint \(a\) and for every \(\epsilon > 0\), there is a \(\delta > 0\) (which in general depends on \(\epsilon, f, I\) and \(a\)) such that
    $$ a - \delta \in I \quad \text{ and } \quad a - \delta < x < a \quad \text{ imply } \quad |f(x) - L| < \epsilon. $$
    In this case we call \(L\) the left-hand limit of \(f\) at \(a\) and denote it by
    $$ \lim_{x \to a-} f(x) = L. $$
Theorem
\(\lim_{x \to a} f(x) = L\) if and only if
$$ \lim_{x \to a-} f(x) = \lim_{x \to a+} f(x) = L. $$

References