Thomas' Calculus 14e — Section 2.1–2.6 Vocabulary (Limits and Continuity)

Vocabulary flashcards covering key concepts from limits and continuity (one-sided limits, limit laws, polynomials, rational functions, continuity, asymptotes, and classic theorems).

Limit (lim f(x) as x→c)

The value L that f(x) approaches as x approaches c, denoted lim x→c f(x) = L. Formal definition: for every ε>0 there exists δ>0 such that 0<|x−c|<δ implies |f(x)−L|<ε.

Right-hand limit

The limit of f(x) as x approaches c from the right: lim x→c+ f(x) = L, considering x > c.

Left-hand limit

The limit of f(x) as x approaches c from the left: lim x→c− f(x) = L, considering x < c.

Two-sided limit

The limit lim x→c f(x) exists if both the left-hand and right-hand limits exist and are equal to the same value L.

Continuity at a point

A function f is continuous at c if f(c) exists and lim x→c f(x) = f(c).

Right-continuous

A function is continuous from the right at c if lim x→c+ f(x) = f(c).

Left-continuous

A function is continuous from the left at c if lim x→c− f(x) = f(c).

Epsilon–delta limit definition

Definition of limit: for every ε>0 there exists δ>0 such that |f(x)−L|<ε whenever 0<|x−c|<δ.

Sandwich (Squeeze) Theorem

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.

Limit laws (Theorem 1) – Sum Rule

If lim f(x) = L and lim g(x) = M, then lim [f(x)+g(x)] = L+M.

Limit laws – Difference Rule

If lim f(x) = L and lim g(x) = M, then lim [f(x)−g(x)] = L−M.

Limit laws – Constant multiple Rule

If lim f(x) = L, then lim [k·f(x)] = k·L for any constant k.

Limit laws – Product Rule

If lim f(x) = L and lim g(x) = M, then lim [f(x)·g(x)] = L·M.

Limit laws – Quotient Rule

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

Limit laws – Power Rule

If lim f(x) = L and n is a positive integer, then lim [f(x)]^n = L^n.

Limit laws – Root Rule

If lim f(x) = L and L ≥ 0, then lim [√(f(x))] = √L.

Polynomials – Limits of polynomials

If P(x) is a polynomial, then lim x→c P(x) = P(c).

Rational functions – Limits

If P and Q are polynomials and Q(c) ≠ 0, then lim x→c [P(x)/Q(x)] = P(c)/Q(c).

Secant line

Line through points P(x1, f(x1)) and Q(x2, f(x2)); its slope is the average rate of change (f(x2)−f(x1))/(x2−x1).

Average rate of change

Slope of the secant line over [x1, x2]: (f(x2)−f(x1))/(x2−x1).

Tangent line

Line through P whose slope is the limit of secant slopes as Q→P; the tangent slope is f′(x) if it exists.

One-sided limits notation reminder

Right-hand limit uses a superscript + (lim x→c+ f(x)); left-hand limit uses a superscript − (lim x→c− f(x)).

Limit of sin x over x (in radians)

As x→0, sin x / x → 1. This is a fundamental trigonometric limit.

Continuity properties (Theorem 8)

If f and g are continuous at c, then sums, differences, constant multiples, products, quotients (where denominator not zero), powers, and roots are continuous at c.

Composition of continuous functions (Theorem 9)

If f is continuous at c and g is continuous at f(c), then the composition g∘f is continuous at c.

Limits of continuous functions (Theorem 10)

If lim x→a f(x) = b and g is continuous at b, then lim x→a g(f(x)) = g(b).

Intermediate Value Theorem (Theorem 11)

If f is continuous on [a,b] and yo is between f(a) and f(b), then there exists c∈[a,b] with f(c) = yo.

Greatest integer function (floor function)

y = [x], continuous at non-integers; right-continuous at integers, but not left-continuous.

Limit at infinity

lim x→∞ f(x) = L means for every ε>0 there exists M such that |f(x)−L|

Horizontal asymptote

A horizontal line y = b is an asymptote if lim x→±∞ f(x) = b.

Vertical asymptote

A vertical line x = a is an asymptote if lim x→a f(x) = ±∞.

Removable discontinuity

A hole in the graph at x = c where the limit exists but f(c) is not equal to that limit (or is undefined at c).