# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: - **One-Sided Limits**: Determining the Left-Hand Limit (LHL) and Right-Hand Limit (RHL) for piecewise functions. - **Continuity at a Point**: A function $f(x)$ is continuous at $x = c$ if the limit exists and equals the function value. - **Finding Unknown Constants**: Using the definition of continuity to solve for variables like $k, m,$ or $n$. - **Points of Discontinuity**: Identifying values of $x$ where the function fails to meet continuity criteria. - **Graphical Analysis**: Understanding how functions behave visually. Understanding these concepts is crucial for solving the exercise questions effectively. --- # **Important Formulas** Below are the key formulas used in this exercise: - **Existence of a Limit**: $$\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$$ - **Definition of Continuity**: A function $f$ is continuous at $c$ if: 1. $f(c)$ is defined. 2. $\lim_{x \to c} f(x)$ exists. 3. $\lim_{x \to c} f(x) = f(c)$. --- # **Summary** This exercise covers the fundamental principles of limits and continuity. Key learnings include evaluating one-sided limits and applying the three-step continuity test to piecewise and rational functions. Mastery involves correctly setting LHL equal to RHL to solve for unknown constants and identifying domain restrictions that lead to discontinuities.