Limits at Infinity for Polynomials and Rational Functions

Overview

This lecture covers techniques for finding limits as x approaches infinity (and negative infinity) for polynomial and rational functions, focusing on how to identify dominant terms and determine the behavior of the function.

Limits with Polynomials at Infinity

• As x approaches infinity, x² grows without bound (limit is infinity).
• As x approaches negative infinity, x² still grows without bound (limit is infinity).
• For x³, as x approaches infinity, limit is infinity; as x approaches negative infinity, limit is negative infinity.
• In polynomials, the highest degree term dominates as x approaches infinity or negative infinity.

Evaluating Limits with Polynomials

• For limits at infinity, ignore lower degree (insignificant) terms in polynomials.
• Example: Limit as x→–∞ of 5+2x–x³ is equivalent to limit of –x³, which gives infinity.
• Example: Limit as x→–∞ of 3x³–5x⁴ is dominated by –5x⁴, so limit is negative infinity.

Limits with Rational Functions at Infinity

• If denominator’s degree > numerator’s, the limit is zero (bottom heavy).
• As x gets large in 1/x, value approaches zero.
• Example: Limit as x→∞ of (5x+2)/(7x–x²) is zero (bottom heavy).

Rational Functions with Equal Degrees

• If numerator and denominator have the same degree, limit is ratio of leading coefficients.
• Example: Limit as x→∞ of (8x²–5x)/(4x²+7) is 8/4 = 2.
• Example: Limit as x→–∞ of (5x–7x³)/(2x²+14x³–9) is (–7)/(14) = –½.

Rational Functions with Higher Degree Numerator (Top Heavy)

• If numerator’s degree > denominator’s, limit is infinity or negative infinity (depends on sign).
• Example: Limit as x→∞ of (5x+6x²)/(3x–8) is infinity.
• Example: Limit as x→–∞ of (5+2x–3x³)/(4x²+9x–7) is infinity (after simplifying dominant terms).

Key Terms & Definitions

• Limit at Infinity — Value a function approaches as x increases or decreases without bound.
• Dominant Term — The term with the highest power of x, controlling behavior as x→∞ or x→–∞.
• Bottom Heavy — Rational function where denominator degree > numerator; limit approaches zero.
• Top Heavy — Rational function where numerator degree > denominator; limit approaches infinity or negative infinity.
• Equal Degree — Rational function where numerator and denominator have same degree; limit is ratio of leading coefficients.

Action Items / Next Steps

• Practice problems: Find limits at infinity for various polynomial and rational functions.
• Review dominant terms for given expressions.