1. **Problem:** Under what condition does the limit $\lim_{x \to a} f(x)$ exist and equal $L$? 2. **Formula and rule:** The limit $\lim_{x \to a} f(x) = L$ exists if and only if the left-hand limit and right-hand limit both exist and are equal to $L$. That is, $$\lim_{x \to a^-} f(x) = L \quad \text{and} \quad \lim_{x \to a^+} f(x) = L.$$ This means the function approaches the same value $L$ from both sides of $a$. 3. **Explanation:** The function value at $a$, $f(a)$, does not necessarily have to be defined or equal to $L$ for the limit to exist. Also, if the function approaches different values from the left and right, the limit does not exist. 4. **Answer:** The correct condition is option C: The left-hand limit $\lim_{x \to a^-} f(x) = L$ and the right-hand limit $\lim_{x \to a^+} f(x) = L$. **Final answer:** C