Mastering AP Calculus AB: Unit 1 Limits Guide

Limit

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

Two-Sided Limit

Exists if a function approaches the same height from both the left and right of a point.

Left-Hand Limit

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

Right-Hand Limit

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

DNE

Does Not Exist; a term used when limits from left and right approach different values.

Continuous Path

When tracing a graph, if the fingers from both sides meet at the same point, the limit exists.

Removable Discontinuity

A situation where the limit exists at a point despite the function being undefined there.

Vertical Asymptote

A vertical line where the function approaches infinity or negative infinity.

Sum/Difference Rule

The limit of the sum/difference of functions equals the sum/difference of their limits.

Product Rule

The limit of the product of functions equals the product of their limits.

Quotient Rule

The limit of the quotient of functions equals the quotient of their limits, provided the denominator is non-zero.

Direct Substitution

A method of evaluating limits by plugging in the value directly into the function.

Indeterminate Form

A limit form that requires further analysis, such as 0/0.

Factoring and Canceling

A technique used to resolve indeterminate forms by simplifying the expression.

Rationalization

A method involving multiplying by the conjugate to eliminate radicals in limit problems.

Special Trigonometric Limits

Limits of the forms sin(x)/x and (1-cos(x))/x at x approaching 0.

Squeeze Theorem

A theorem used to find limits of functions trapped between two other functions that converge to the same limit.

Graphical Approach

Estimating limits by visually tracing a function on its graph.

Numerical Approach

Estimating limits by analyzing values in a table as they approach a specified point.

Limit Notation

Expressed as lim x → c f(x) = L, where L is the limit as x approaches c.

Behavior of Limits at Infinity

Describing how limits behave as x approaches infinity or negative infinity.

One-Sided Limit Notation

Written as lim x → c^- and lim x → c^+ for left-hand and right-hand limits, respectively.

Continuous Function

A function that is continuous at a point if the limit equals the value of the function at that point.

Hole

A point of discontinuity where the function is not defined but where the limit exists.

Jump Discontinuity

A point where the function's limits from the left and right do not match.

Asymptotic Behavior

The behavior of functions as they approach vertical or horizontal lines in a limit.

The Existence Theorem

States that a two-sided limit exists if the left-hand and right-hand limits both equal the same value.