Overview
This lecture introduces limits at infinity, including how functions behave as x approaches infinity or negative infinity, and methods for evaluating these limits with polynomials and rational functions.
Limits at Infinity: Concepts and Definitions
- Limits at infinity analyze what happens to f(x) as x becomes very large (positive or negative).
- The limit as x approaches infinity (or negative infinity) of f(x) can be a real number, infinity, or may not exist.
- A function approaches a horizontal asymptote y = L if the limit as x approaches infinity or negative infinity of f(x) is L.
Formal Definition and Example
- To show the limit of 1/x² as x approaches infinity is 0, show |1/x² – 0| < ε for any ε > 0 with sufficiently large x.
- For any small ε > 0, choosing x > 1/√ε ensures |1/x²| < ε.
Graphical Implications: Horizontal Asymptotes
- If the limit as x approaches infinity of f(x) is L, y = L is a horizontal asymptote.
- Horizontal asymptotes may differ for x approaching infinity versus negative infinity.
- There cannot be two distinct asymptotes in the same direction; specify direction for each.
Power Functions and Roots
- If n > 0, lim_{x→∞} xⁿ = ∞; if n < 0, lim_{x→∞} xⁿ = 0.
- For negative infinity and rational exponents, existence depends on whether roots are even or odd.
- Even roots of negative numbers (like 4th root of x when x < 0) are undefined.
Evaluating Limits: Examples
- Multiply by negative constants to flip sign; e.g., lim_{x→∞} (–6x³) = –∞.
- 1/x^positive power goes to 0 as x approaches infinity.
- Cube roots of negative x are defined, but even roots of negative x are not.
Polynomials and Dominant Terms
- As x approaches infinity or negative infinity, only the highest power term matters.
- For polynomials, if the highest power's coefficient is positive, the limit is ∞; negative, the limit is –∞.
- If the dominant term is undefined for given x (like even roots of negatives), the limit does not exist.
Rational Functions
- For lim_{x→∞} (polynomial/polynomial), compare degrees (highest powers) in numerator (m) and denominator (n):
- If m < n: limit is 0 (horizontal asymptote y = 0).
- If m = n: limit is ratio of leading coefficients (horizontal asymptote y = coefficient ratio).
- If m > n: limit is ±∞; sign depends on leading coefficients.
- The same rules apply for x approaching negative infinity, but pay close attention to sign changes.
Key Terms & Definitions
- Limit at Infinity — Value f(x) approaches as x increases or decreases without bound.
- Horizontal Asymptote — Line y = L where f(x) approaches L as x → ∞ or x → –∞.
- Dominant Term — The term with the highest power in a polynomial affecting end-behavior.
- Rational Function — A ratio of two polynomials.
- Epsilon (ε) — A small positive number used in formal limit definitions.
Action Items / Next Steps
- Practice finding limits at infinity for various polynomials and rational functions.
- Review the formal epsilon-delta definition of limits.
- Read the textbook section on horizontal asymptotes and rational functions.