BackLimits at Infinity and Horizontal Asymptotes 2.5
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Limits at Infinity
Limits at Infinity of Polynomials
When evaluating the limit of a polynomial function as approaches infinity or negative infinity, the term with the highest degree dominates the behavior of the function.
General Form: , where
Limit as :
Limit as :
Example:
Limits at Infinity of Rational Functions
For rational functions, the behavior as approaches infinity depends on the degrees of the numerator and denominator polynomials.
Case 1: Degree of numerator < degree of denominator: Limit is 0.
Case 2: Degree of numerator = degree of denominator: Limit is the ratio of leading coefficients.
Case 3: Degree of numerator > degree of denominator: Limit is (does not exist).
Examples:
Limits at Infinity of Functions with Negative Powers
Functions with terms like , where , approach zero as approaches infinity.
Example:
Horizontal Asymptotes (H.A.)
Definition and Identification
A horizontal asymptote of a function is a horizontal line that the graph of $f(x)$ approaches as tends to or .
General Form:
Rules for Finding Horizontal Asymptotes
Case 1: Degree of < degree of Horizontal Asymptote: Example: H.A.:
Case 2: Degree of = degree of Horizontal Asymptote: Example: H.A.:
Case 3: Degree of > degree of Horizontal Asymptote: Does Not Exist (DNE) Example: H.A.: DNE
Example: Finding Vertical and Horizontal Asymptotes
Consider
Find the Vertical Asymptote (V.A.): Factor numerator: (for ) Vertical asymptote at
Identify the Horizontal Asymptote (H.A.): Leading coefficients: $1 (denominator)
Summary Table: Horizontal Asymptotes of Rational Functions
Degree of Numerator | Degree of Denominator | Horizontal Asymptote | Example |
|---|---|---|---|
< Denominator | Higher | ||
= Denominator | Equal | ||
> Denominator | Lower | DNE |
Key Terms
Limit at Infinity: The value a function approaches as increases or decreases without bound.
Horizontal Asymptote: A horizontal line that the graph of a function approaches as or .
Vertical Asymptote: A vertical line where the function grows without bound as approaches .
Additional info:
All examples and explanations are based on standard calculus concepts for limits at infinity and asymptotes.
Graphs referenced in the notes illustrate the behavior of rational functions near their asymptotes.
