AP Calculus AB Unit 1 Study Guide: Limits Involving Infinity (Asymptotes) + Intermediate Value Theorem

Infinite Limit

Describes a function whose values grow without bound as the input approaches a finite number.

Vertical Asymptote

A vertical line where the graph approaches and the function values become unbounded as x approaches a certain value.

Limit Notation

Mathematical notation used to express limits, such as $\lim_{x \to a} f(x)$.

DNE

Does Not Exist; used when the limit fails to approach a finite number.

One-sided Infinite Limit

Limits that describe behavior approaching a number from one side (left or right), e.g., $\lim_{x \to a^-} f(x)$.

Vertical Blow-up Behavior

The behavior of a function near a vertical asymptote where it goes to infinity or negative infinity.

Candidates for Vertical Asymptotes

Points where the denominator of a rational function equals zero and the numerator does not.

Non-Zero over Zero Rule

A rule stating that a vertical asymptote occurs at x=c if the denominator at c is zero and the numerator is nonzero.

Sign Analysis

A method used to determine the direction of the function's values as they approach a vertical asymptote.

Horizontal Asymptote

A horizontal line that describes the end behavior of a function as x approaches infinity or negative infinity.

Case 1: Degree Comparison

If the degree of the numerator is less than the degree of the denominator, the limit approaches zero.

Case 2: Equal Degree

If the degrees of both the numerator and the denominator are equal, the limit equals the ratio of the leading coefficients.

Case 3: Degree Comparison (Top Heavy)

If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

BOBO BOTN EATS DC

Mnemonic summarizing rational function degree cases: Bottom Heavy, Zero; Bigger On Top, None; Equal degrees, Average Coefficients.

Dominance Order

A ranking of function types by growth rate: Exponentials > Polynomials > Logarithms > Bounded functions.

Intermediate Value Theorem (IVT)

A theorem stating that a continuous function on a closed interval must hit any value between its output endpoints.

Continuity Requirement for IVT

A function must be continuous on an interval to apply the Intermediate Value Theorem.

Existence Conclusion of IVT

The conclusion that at least one c exists in an interval satisfying f(c) = N, based on the IVT.

Endpoint Evaluation

Calculating the values of a function at the endpoints of an interval to apply the IVT.

Sign Change Condition

A condition that is often checked when using IVT to guarantee the existence of a root between two points.

Non-Applicability of IVT

IVT cannot be used if the function is not continuous on the specified interval.

Removable Discontinuity

A point where a function is undefined but can be defined to make the function continuous.

Indeterminate Form

A mathematical condition, such as $\frac{0}{0}$, indicating a limit requires further evaluation.

Graph Interpretation of IVT

Using the graph of a continuous function to visually confirm that it meets the value conditions of the IVT.

Rational Function

A function defined by the ratio of two polynomials.

Asymptotic Behavior

The behavior of a function as it approaches a vertical or horizontal asymptote.

Limit at Infinity

Describes what happens to a function as the variable approaches infinity or negative infinity.