Video Notes: Determining Limits of Trigonometric Functions and Identities

Vocabulary flashcards covering key concepts from limits of trigonometric functions, including indeterminate forms, substitution, simplification techniques, and essential trigonometric identities.

Limit (calculus)

The value that a function or expression approaches as the input variable approaches a specified point.

Indeterminate form 0/0

A form that occurs when substitution yields 0 in the numerator and 0 in the denominator, requiring further simplification or methods to evaluate the limit.

Indeterminate form ∞/0

A form that signals the limit cannot be directly determined and requires algebraic manipulation or identities to evaluate.

Substitution method in limits

Replacing the variable with the approaching value to find the limit, which may fail if the result is an indeterminate form.

Algebraic simplification in limits

Using algebraic techniques and identities to rewrite the expression so that the limit can be evaluated.

Trigonometric identities

Equations that relate trigonometric functions to each other and to constants, used to simplify limits and expressions.

Pythagorean identities

Key relations: sin^2 x + cos^2 x = 1; tan^2 x + 1 = sec^2 x; 1 + cot^2 x = csc^2 x.

Reciprocal identities

Relationships: sin x and csc x; cos x and sec x; tan x and cot x (e.g., csc x = 1/sin x, sec x = 1/cos x, cot x = cos x / sin x).

Quotient identities

Definitions of tangent and cotangent in terms of sine and cosine: tan x = sin x / cos x; cot x = cos x / sin x.

Double-angle identities

Formulas for double angles: sin 2x = 2 sin x cos x; cos 2x = cos^2 x − sin^2 x (also cos 2x = 1 − 2 sin^2 x, or cos 2x = 2 cos^2 x − 1).

Factoring in trigonometric limits

Using factorization such as cos^2 x − 1 = (cos x − 1)(cos x + 1) to cancel terms and simplify a limit.

Cancellation via common factors

Canceling common factors that appear in numerator and denominator to resolve a limit (e.g., removing a (cos x + 1) factor).

Identity sin^2 x + cos^2 x = 1

A fundamental Pythagorean identity relating sine and cosine.

Identity sin 2x = 2 sin x cos x

Double-angle identity for sine used to simplify expressions involving sin of a double angle.

Identity cos 2x = cos^2 x − sin^2 x

Double-angle identity for cosine, with alternative forms cos 2x = 1 − 2 sin^2 x and cos 2x = 2 cos^2 x − 1.

Identity tan^2 x + 1 = sec^2 x

Pythagorean identity connecting tangent and secant functions.

Identity 1 + cot^2 x = csc^2 x

Pythagorean identity connecting cotangent and cosecant functions.