Limit and continuity applications include: (1) Existence of solutions (IVT): If f continuous on [a,b] and f(a)·f(b) < 0, at least one root exists; (2) Function behavior: Analyze growth, decay, asymptotes; (3) Continuity in real-world models: Physical quantities change continuously. Example: Temperature must pass through all intermediate values (IVT). Using limits: Find where function approaches value without reaching it (horizontal/vertical asymptotes). Solving: Apply theorems appropriately to guarantee solutions. Graph to visualize behavior. Exam tip: Use IVT to establish root existence. Understand continuous functions preserve intervals and have intermediate values.