Undefined limits occur when the limit approaches infinity, as in the case of \lim_{x \to 0} \frac{1}{x^2}, where both sides converge to infinity but do not yield a real number. In contrast, limits that do not exist, such as \lim_{x \to 0} \frac{1}{x}, arise when the left-hand and right-hand limits differ. The distinction lies in the behavior of the limits: undefined limits approach infinity, while non-existent limits fail to converge to a single value. Understanding these differences is crucial for solving calculus problems effectively. Clarifying these concepts can significantly enhance comprehension of limits in mathematical analysis.