Calculus Volume 1 – Chapter 2: Limits (Vocabulary Flashcards)

Limit

The value L that f(x) approaches as x approaches a; symbolically lim_{x→a} f(x) = L; intuition: as x gets near a, f(x) gets near L.

One-Sided Limit (Left)

The limit as x approaches a from the left: lim_{x→a^-} f(x) = L.

One-Sided Limit (Right)

The limit as x approaches a from the right: lim_{x→a^+} f(x) = L.

Two-Sided Limit

If both one-sided limits exist and are equal, then lim_{x→a} f(x) = L.

Infinite Limit

A limit that grows without bound, i.e., tends to ∞ or -∞ as x approaches a.

Vertical Asymptote

A vertical line x = a where f(x) increases without bound as x → a.

Does Not Exist (DNE)

A limit that does not exist; may be due to differing left/right limits or unbounded behavior.

Limit Laws – Quotient Law

If lim f(x) and lim g(x) exist and lim g(x) ≠ 0, then lim f(x)/g(x) = [lim f(x)]/[lim g(x)].

Limit Laws – Power Law

lim [f(x)]^n = [lim f(x)]^n for positive integer n, when the limit exists.

Limit Laws – Root Law

lim [f(x)]^{1/n} = [lim f(x)]^{1/n} under appropriate conditions.

Tangent Problem

Problem of finding instantaneous rate of change; the slope of the tangent line equals the instantaneous rate of change, defined as a limit.

Secant Line

Line through two points on a graph; slope m = [f(x+h) - f(x)]/h; as h → 0, secant slope approaches the tangent slope.

Tangent Line

Line that best approximates the function at a point; slope equals the derivative (when the limit exists).

Instantaneous Rate of Change

Rate of change at a single point; a limit of the average rate of change as the interval tends to zero.

Average Rate of Change

Rate of change between two points: [f(b) - f(a)]/(b - a).

Instantaneous Velocity

v(a) = lim_{h→0} [s(a+h) - s(a)]/h; instantaneous rate of motion at time a.

Position Function

s(t): position of an object at time t.

Area Problem

Finding the area under a curve; leads to integral calculus; approximate with rectangles and refine to exact area.

Riemann Sum

Sum of rectangle areas used to approximate the