Overview
This lecture covers limits at infinity, focusing on rational and related functions. It explains algebraic techniques to determine asymptotic behavior as x approaches infinity or negative infinity.
Limits of Reciprocal Powers
- As x → ∞, 1/x, 1/x², and 1/√x all approach zero because the denominator grows without bound.
- 1/x² approaches zero faster than 1/x, and 1/√x approaches zero slower than 1/x.
- For any positive rational r, limₓ→∞ 1/xʳ = 0.
- If xʳ is defined for all x, limₓ→−∞ 1/xʳ = 0 (works for odd roots, not even roots of negatives).
Evaluating Rational Function Limits
- For a rational function, both numerator and denominator approach infinity as x → ∞.
- To analyze the limit, divide numerator and denominator by the highest power of x in the denominator.
- Example: limₓ→∞ (5x² + 2x − 3)/(3x² + 9x) = 5/3 after dividing by x².
- As x → −∞, similar steps yield limits (e.g., for linear terms, divide by x).
Special Cases in Rational Functions
- If numerator and denominator have the same degree, limit is the ratio of leading coefficients.
- If numerator degree < denominator degree, limit is zero.
- If numerator degree > denominator degree, not covered here (to be discussed later).
- Always perform limit calculations; don't rely solely on degree comparison.
Horizontal Asymptotes and Degree Comparison
- Horizontal asymptote at y = 0 if numerator degree < denominator.
- Horizontal asymptote at ratio of leading coefficients if degrees are equal.
- Asymptote must be verified with limit calculations, not just by observation.
Key Terms & Definitions
- Limit at infinity — The value a function approaches as x becomes very large or very small.
- Rational function — A function that is the ratio of two polynomials.
- Horizontal asymptote — A horizontal line that the graph of a function approaches as x → ±∞.
Action Items / Next Steps
- Practice checking limits of rational functions using division by highest degree x.
- Prepare for cases where numerator degree > denominator degree in future lessons.