AP Calculus AB Unit 1 (Limits): Notation, Estimation, Algebraic Techniques, Strategy, and the Squeeze Theorem

Limit

A value that a function approaches as the input approaches a specified value.

Two-sided limit

The value that a function approaches as the input approaches a specified point from both the left and right.

One-sided limit

The value that a function approaches as the input approaches a specified point from one direction only.

Existence Theorem

States that a two-sided limit exists if and only if the left-hand limit equals the right-hand limit.

DNE (Does Not Exist)

Indicates that a limit does not approach a single value.

Continuous function

A function where limits can be directly substituted at any point within its domain.

Indeterminate form

An expression like 0/0 that requires further analysis for limit determination.

Squeeze Theorem

States that if a function is trapped between two others that converge to the same limit, it must also converge to that limit.

Infinite limit

A scenario where a function grows without bound as the input approaches a particular value.

Jump discontinuity

A situation in which a function has different limits when approached from different sides.

Oscillation

When a function continuously bounces between values without settling as the input approaches a limit.

Factoring

The algebraic process of breaking down a polynomial into simpler components to simplify a limit calculation.

Rationalizing

A technique used to eliminate radicals from the denominator by multiplying by a conjugate.

L'Hôpital's Rule

A method for evaluating limits of indeterminate forms by taking derivatives of the numerator and denominator.

Trigonometric limit

Limits involving trigonometric functions, often utilizing known limits like sin(x)/x as x approaches 0.

Left-hand limit

The limit of a function as the input approaches a specified value from the left.

Right-hand limit

The limit of a function as the input approaches a specified value from the right.

Limit notation

Expressed as lim_{x -> a} f(x) = L, indicating how f(x) behaves as x approaches 'a'.

Vertical asymptote

A line that a function approaches as it increases or decreases without bound.

Substitution method

An initial limit evaluation technique where a function is directly evaluated at a specified input.

Continuous path

A scenario in which the function behaves smoothly without interruptions or gaps.

Piecewise function

A function defined by multiple sub-functions, each applying to a specific interval of the input.

Average rate of change

The change in the function's output divided by the change in the input over an interval.

Riemann Sum

An approximation of integrals, formed by summing products of function values at certain points and intervals.

Jump (discontinuity)

A point where a function abruptly changes from one value to another, creating a gap.