Unit 2 Summary & Key Concepts

AP Calculus · Axiom Academy

SUMMARY Unit 2: Limits & Continuity Review of all key concepts and formulas Limits are the foundation of calculus. They describe how functions behave near (not necessarily at) specific points. means f(x) approaches L as x approaches a from both sides. Key insight: The function might not equal L at x = a, and it might not even be defined there! Two-sided limit exists ⟺ left limit = right limit For continuous functions, direct substitution works. For indeterminate forms, factor, use conjugates, or apply special limits. f is continuous at x = a if three conditions hold: Removable: Hole (limit exists but ≠ f(a)) Jump: Left and right limits differ Oscillating: Function bounces wildly The Intermediate Value Theorem If f is continuous on [a, b] and M is between f(a) and f(b), then for some c in (a, b). Use this to prove roots exist without finding them! For rational functions, the degree of numerator vs. denominator determines the horizontal asymptote: degree(N) > degree(D): No horizontal asymptote Defining derivatives (rate of change) Defining integrals (accumulation) Proving continuity and applying theorems Understanding asymptotic behavior Real-world applications (velocity, growth rates, etc.) Now that you master limits and continuity, you're ready for Unit 3: The Derivative!

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