AP Calculus AB Unit 1: Limits and Continuity (Comprehensive Study Guide)

Limit

A value that a function's output approaches as the input approaches a specific value.

Two-sided limit

The limit that evaluates what happens as you approach a specific value from both sides.

Left-hand limit

The limit that evaluates what happens as you approach a specific value from the left.

Right-hand limit

The limit that evaluates what happens as you approach a specific value from the right.

Indeterminate form

A form like 0/0 that does not provide enough information to determine a limit.

Removable discontinuity

A discontinuity where the limit exists but the function value is either missing or does not equal the limit.

Jump discontinuity

A discontinuity where the left-hand limit and right-hand limit exist but are not equal.

Infinite discontinuity

Occurs when at least one of the one-sided limits approaches infinity.

Continuity at a point

A function is continuous at a point if it is defined at that point, the limit exists, and the limit equals the function value.

Intermediate Value Theorem (IVT)

The theorem stating that if a function is continuous on an interval, then it takes on every value between f(a) and f(b).

Tend towards

To approach a specific value as another value changes.

Asymptote

A line that a curve approaches as it heads towards infinity.

Vertical asymptote

A vertical line that represents a value that a function approaches as x approaches a specific value.

Horizontal asymptote

A horizontal line that represents the end behavior of a function as x approaches infinity.

Limit laws

Rules that describe how limits can be computed using algebraic operations.

Squeeze theorem

A theorem that states if f(x) is squeezed between g(x) and h(x), and both g(x) and h(x) approach the same limit L, then f(x) must also approach L.

Direct substitution

Plugging the value directly into the function to evaluate a limit.

Nonexistent limit

A situation where a limit cannot be determined or does not approach a single finite value.

Continuous function

A function that has no breaks, jumps, or holes across its entire domain.

Function value

The output of a function for a given input.

Limit value

The value that a function approaches as the input approaches a specific value.

End behavior

How a function behaves as the input approaches infinity.

Polynomial continuity

Polynomials are continuous everywhere in their domain.

Rational function discontinuity

Rational functions are continuous except where the denominator is zero.

Testing continuity

To determine continuity at a point, check if the function is defined, if the limit exists, and if they are equal.

Conjugate multiplication

A method used to eliminate square roots in the denominator by multiplying by the conjugate.

Sign change

Occurs when the function value differs in sign at two points, indicating a zero exists between these points.

Function hole

An x-value that is not included in the function due to a removable discontinuity, often seen as an open circle in graphs.

Behavior near

Understanding how a function acts in the vicinity of a certain value.

Graph estimation

Using visual representation of graphs to estimate limits at specific points.

Limit definition

The formal definition used to express the approach of f(x) to a limit L as x approaches a.

Piecewise function

A function that is defined by different expressions depending on the input value.

Rate of change

How a quantity changes in relation to another quantity, often expressed using derivatives.

Oscillation behavior

When the values of a function fluctuate rapidly without approaching a single limit.

Common denominator

A common base used when adding or subtracting fractions; helps simplify rational functions.

End behavior model

Mathematical model to describe how functions behave as x approaches positive or negative infinity.

Numerical table limit estimation

Using a numerical table to approach a limit rather than traditional substitution.

Limit existence theorem

A theorem stating that a two-sided limit exists if and only if both one-sided limits exist and are equal.

Continuous output

The outputs of a function that continuously vary without jumps or interruptions.

Converging values

When values approach a single point or value.

Function graph smoothness

A characteristic of a function's graph that indicates it has no sharp corners or breaks.

Trigonometric limits

Limits involving trigonometric functions that have specific known behaviors.

Vertical growth behavior

The behavior of functions that head towards infinity or negative infinity near a specific point.

Approaching defined value

The scenario in which a result closes in on a specific value without necessarily reaching it.

Discontinuity types

The classification of discontinuities into removable, jump, and infinite.

Interpreting limit notation

Understanding what limit notation implies regarding approaching behavior rather than direct equality.