Overview
This lecture explains how to find the limits of reciprocal functions analytically, emphasizing how the function's coefficient and denominator's degree impact end behavior and limits near zero.
Structure of Reciprocal Functions
- A reciprocal function has the form c / xⁿ, where c ≠ 0 and n is a natural number.
- The coefficient c can be positive or negative, and n can be even or odd.
- Negative coefficients are often factored out and applied to the entire fraction.
Special Properties and Behaviors
- All reciprocal functions share similar end behaviors and only four possible behaviors around zero.
- The behavior near zero depends on the signs of the coefficient and whether n is even or odd.
- For x ≠ 0, reciprocal functions are continuous, so the limit at any nonzero point is just the function's value.
- As x → ±∞, all reciprocal functions approach zero.
Limits Based on Coefficient and Degree
- Positive c, odd n:
- As x → 0⁻, f(x) → −∞.
- As x → 0⁺, f(x) → +∞.
- Positive c, even n:
- As x → 0⁻, f(x) → +∞.
- As x → 0⁺, f(x) → +∞.
- Negative c, odd n:
- As x → 0⁻, f(x) → +∞.
- As x → 0⁺, f(x) → −∞.
- Negative c, even n:
- As x → 0⁻, f(x) → −∞.
- As x → 0⁺, f(x) → −∞.
Calculating Limits Analytically
- Identify the coefficient's sign and degree's parity to choose the right limit behavior.
- For further understanding, sketch the graph to visualize limit behaviors.
- For all x ≠ 0, the limit is simply the function's value due to continuity.
Key Terms & Definitions
- Reciprocal Function — A function of the form c / xⁿ with c ≠ 0 and n a natural number.
- Coefficient (c) — The non-zero number multiplying the fraction.
- Degree (n) — The natural number exponent in the denominator.
- Odd Degree — n is odd (e.g., 1, 3, 5), affects sign changes.
- Even Degree — n is even (e.g., 2, 4, 6), produces same-sign behavior on both sides of zero.
- Continuous — The function has no breaks except possibly at x = 0.
Action Items / Next Steps
- Practice associating reciprocal functions with their correct limit statements using coefficient sign and degree parity.
- Sketch quick graphs of reciprocal functions to predict limit behavior.
- Review provided chart or summary for the four possible behaviors near zero.