Limits and Continuity: every key term you need

Limits and Continuity

Limit: value function approaches as input approaches some value. Continuity: function is unbroken at a point; requires limit exists and equals function value.

Limit Definition

lim(x→a) f(x) = L means for every ε > 0, there exists δ > 0 such that |x-a| < δ implies |f(x)-L| < ε; rigorous definition of limit.

One-Sided Limits

Left limit: lim(x→a⁻) f(x); right limit: lim(x→a⁺) f(x). Two-sided limit exists only if both one-sided limits exist and are equal.

Limit Laws

Sum, product, quotient, power rules; if lim(x→a) f(x) = L and lim(x→a) g(x) = M, then lim(x→a) [f(x)+g(x)] = L+M (and similar for other operations).

Infinite Limits

lim(x→a) f(x) = ∞ means function grows unbounded; often indicates vertical asymptote at x = a.

Limits at Infinity

lim(x→∞) f(x) = L means function approaches L as x grows; describes horizontal asymptotes; key for analyzing end behavior.

Squeeze Theorem

If g(x) ≤ f(x) ≤ h(x) near a, and lim(x→a) g(x) = lim(x→a) h(x) = L, then lim(x→a) f(x) = L; useful for hard-to-evaluate limits.

Continuity at a Point

f is continuous at a if lim(x→a) f(x) = f(a); three conditions: limit exists, function defined at a, and they're equal.

Types of Discontinuities

Removable: hole (limit exists, f(a) undefined or wrong); jump: left/right limits differ; infinite: vertical asymptote.

Intermediate Value Theorem

If f is continuous on [a,b] and N between f(a) and f(b), then there exists c in (a,b) where f(c) = N; guarantees solutions exist.

Asymptotes

Vertical: x = a where function undefined/unbounded (lim = ±∞); horizontal: y = L where lim(x→±∞) f(x) = L; oblique: linear asymptote.

Polynomials and Limits

Polynomial limits: substitute directly (no indeterminate form). Rational functions: factor and cancel before substituting to resolve 0/0 form.

Indeterminate Forms

0/0, ∞/∞, 0·∞, ∞-∞, 0⁰, 1^∞, ∞⁰: these are ambiguous; require algebraic manipulation or L'Hôpital's rule to evaluate.

L'Hôpital's Rule (Preview)

For indeterminate forms 0/0 or ∞/∞: lim f(x)/g(x) = lim f'(x)/g'(x); covered in detail in Unit 5.

Trigonometric Limits

Key limit: lim(x→0) sin(x)/x = 1; useful for evaluating trig limit problems; derived using squeeze theorem.

Exponential and Logarithmic Limits

lim(x→∞) (1+1/x)^x = e; lim(x→0⁺) x·ln(x) = 0; key limits for transcendental functions.

Continuity of Basic Functions

Polynomials, rational functions (where defined), trig functions, exponential, logarithm are continuous on their domains.

Composition Continuity

If f continuous at g(a) and g continuous at a, then (f∘g) continuous at a; allows analyzing complex composite functions.

Extreme Value Theorem

If f continuous on [a,b], then f attains max and min on [a,b]; guarantees extrema exist for continuous functions on closed intervals.

Informal Limit Idea

Limit describes behavior of function near (not at) a point; foundation for calculus; different from value of function.

More AP Calculus AB guides

• Derivative Definition
• Extreme Value Theorem
• Antiderivative Definition
• Differential Equations Basics