Unit 1 · Limits and Continuity

Average rate of change ● (core concept) — The change in a function's output divided by the change in its input over an interval: [f(b) − f(a)] / (b − a). It equals the slope of the secant line through the two points.

Instantaneous rate of change ● (core concept) — The rate of change of a function at a single point, defined as the limit of average rates of change over intervals shrinking to that point. It equals the slope of the tangent line, i.e., the derivative.

Limit of a function ● (core concept) — The value L that f(x) approaches as x approaches c: lim(x→c) f(x) = L means f(x) can be made arbitrarily close to L by taking x sufficiently close to c (but not equal to c).

Limit notation ● (core concept) — The symbolic form lim(x→c) f(x) = L, read 'the limit of f of x as x approaches c equals L.' Correct notation is required when justifying limit claims.

One-sided limit ● (core concept) — A limit as x approaches c from one side only: from the right, lim(x→c⁺), or from the left, lim(x→c⁻). The two-sided limit exists only if both one-sided limits exist and are equal.

Algebraic properties of limits (limit laws) ● (core concept) — Rules for combining limits: the limit of a sum, difference, product, quotient, or power equals the corresponding combination of the individual limits, provided each limit exists and the denominator's limit is not zero.

Squeeze Theorem ● (core concept) — If g(x) ≤ f(x) ≤ h(x) near c and lim(x→c) g(x) = lim(x→c) h(x) = L, then lim(x→c) f(x) = L.

Removable discontinuity ● (core concept) — A discontinuity at x = c where the limit exists but either f(c) is undefined or f(c) does not equal the limit; the graph has a hole that could be filled by redefining f(c).

Jump discontinuity ● (core concept) — A discontinuity at x = c where the left- and right-hand limits exist but are not equal, so the graph 'jumps' from one value to another.

Discontinuity due to a vertical asymptote ● (core concept) — A discontinuity at x = c where the function grows without bound (an infinite limit) as x approaches c, producing a vertical asymptote there.

Continuity at a point ● (core concept) — A function f is continuous at x = c if three conditions hold: f(c) is defined, lim(x→c) f(x) exists, and lim(x→c) f(x) = f(c).

Continuity over an interval ● (core concept) — A function is continuous over an interval if it is continuous at every point of the interval; at an endpoint, the appropriate one-sided limit must equal the function value.

Infinite limit ● (core concept) — A limit that grows without bound, written lim(x→c) f(x) = ∞ or −∞; the limit does not exist as a real number.

Vertical asymptote ● (core concept) — The line x = c is a vertical asymptote of f if f has an infinite limit as x approaches c from at least one side.

Limit at infinity ● (core concept) — The behavior of f(x) as x grows without bound: lim(x→∞) f(x) = L means f(x) approaches L for arbitrarily large x.

Horizontal asymptote ● (core concept) — The line y = L is a horizontal asymptote of f if lim(x→∞) f(x) = L or lim(x→−∞) f(x) = L.

End behavior ○ — A description of what happens to f(x) as x → ∞ and as x → −∞, often summarized with horizontal asymptotes.

Intermediate Value Theorem (IVT) ● (core concept) — If f is continuous on [a, b] and N is any value between f(a) and f(b), then there is at least one c in (a, b) with f(c) = N.