Certainly! Below is the transcription and explanation suitable for an educational website: --- **Limit Problems for Calculus Practice** Here are three limit expressions that require evaluation: a. \(\lim_{{x \to 3^+}} \frac{x^2 - 2x - 3}{(x - 3)^3}\) b. \(\lim_{{x \to 3^-}} \frac{x^2 - 2x - 3}{(x - 3)^3}\) c. \(\lim_{{x \to 3}} \frac{x^2 - 2x - 3}{(x - 3)^3}\) ### Explanation: - **Expression a, b, and c** all involve finding the limit of a rational function where the numerator is a quadratic expression \(x^2 - 2x - 3\), and the denominator is the cubic expression \((x - 3)^3\). - **Expression a**: The limit \(\lim_{{x \to 3^+}}...\) means that \(x\) approaches 3 from the right (values greater than 3). - **Expression b**: The limit \(\lim_{{x \to 3^-}}...\) indicates that \(x\) approaches 3 from the left (values less than 3). - **Expression c**: The limit \(\lim_{{x \to 3}}...\) is the two-sided limit as \(x\) approaches 3 from both directions. Calculating these limits involves factoring the quadratic in the numerator, simplifying the expression if possible, and analyzing the behavior as \(x\) approaches 3 from specified directions. This exercise helps understand concepts of one-sided and two-sided limits, particularly when dealing with indeterminate forms. ---