Limits and Continuity Review

Vocabulary flashcards covering key definitions, theorems, and identities related to limits, continuity, and trigonometry.

Precise Definition of a Limit

For every ε > 0, there exists δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Continuity at a Point

f is continuous at a if lim(x→a) f(x) = f(a).

Vertical Asymptote

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

Horizontal Asymptote

y = L is a horizontal asymptote if lim(x→∞) f(x) = L or lim(x→-∞) f(x) = L.

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.

Intermediate Value Theorem

If f is continuous on [a, b], and N is between f(a) and f(b), then there exists c in (a, b) such that f(c) = N.

Special Limit for Sine

lim(t→0) (sin t / t) = 1.

Limits of Elementary Functions

Polynomials/rational: use substitution (if denom ≠ 0). Exponential/logarithmic: use continuity. Trig: use special limits and identities.

Pythagorean Identity

sin²x + cos²x = 1.

Tangent Identity

tan x = sin x / cos x.

Cotangent Identity

cot x = cos x / sin x.

Reciprocal Identities

sec x = 1/cos x, csc x = 1/sin x, cot x = 1/tan x.

Double Angle for Sine

sin(2x) = 2 sin x cos x.

Double Angle for Cosine

cos(2x) = cos²x - sin²x = 2cos²x - 1 = 1 - 2sin²x.

Special Limit for Tangent

lim(x→0) (tan x / x) = 1.

Half-Angle Identities

sin²x = (1 - cos(2x))/2, cos²x = (1 + cos(2x))/2.

Small Angle Approximations

As x→0: sin x ≈ x, tan x ≈ x, cos x ≈ 1 - x²/2.