Limits and Continuity — Quick Review

GRE Math Subject · Axiom Academy

LESSON Limits and Continuity — Quick Review Foundation for all calculus: rigorous definitions and key theorems Step 1: Epsilon-Delta Definition (Formal) lim(x→a) f(x) = L if for every ε > 0, there exists δ > 0 such that: 0 < |x - a| < δ ⟹ |f(x) - L| < ε You probably won't need to use this definition directly on the test, but it underpins all limit behavior. Key insight: A limit exists when f(x) gets arbitrarily close to L as x approaches a. Step 2: Limit Laws and Evaluation If lim(x→a) f(x) = L and lim(x→a) g(x) = M, then: lim(x→a) [f(x) + g(x)] = L + M lim(x→a) [cf(x)] = cL (constant multiple) lim(x→a) [f(x)/g(x)] = L/M (if M ≠ 0) For direct substitution (when function is continuous at a): Indeterminate forms (need L'Hôpital's Rule or algebraic manipulation): 0/0, ∞/∞, 0·∞, ∞-∞, 0^0, 1^∞, ∞^0 If lim(x→a) f(x)/g(x) produces 0/0 or ∞/∞, then: lim(x→a) f(x)/g(x) = lim(x→a) f'(x)/g'(x) (provided the limit on the right exists) Critical: Differentiate numerator and denominator separately. Don't use quotient rule. Common limits you should memorize: Definition: f is continuous at x = a if: Removable: Limit exists but f(a) ≠ limit (or undefined). Can be "fixed". Jump: Left and right limits exist but don't match. Intermediate Value Theorem: If f is continuous on [a,b] and k is between f(a) and f(b), then there exists c ∈ (a,b) where f(c) = k. This is vital for proving roots exist.

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