Understanding Reciprocal Functions and Limits

Sep 6, 2024

Lecture Notes: Reciprocal Functions and Limits at Infinity

Overview

  • Discussion on reciprocal functions and their graphs.
  • Focus on the behavior of these functions as x approaches positive or negative infinity.
  • Importance of understanding these functions for limits at infinity.

Reciprocal Functions with Odd Powers

  • Odd Power Denominator: Graphs have asymptotes.
    • Vertical asymptote at x = 0.
    • Horizontal asymptote at y = 0.
  • Graph Characteristics:
    • Positive x raised to an odd power stays positive.
    • Negative x raised to an odd power stays negative.
    • As x approaches ±∞, the graph approaches y = 0 but never reaches it.

Reciprocal Functions with Even Powers

  • Even Power Denominator:
    • Similar asymptotes as odd power functions.
    • Raising negative x to an even power results in positive y-values.
  • Graph Behavior:
    • As x approaches ±∞, y-values approach zero.
    • Both ends of the graph head towards positive y.

Importance in Class

  • The graph of 1/x^n is crucial for understanding limits at infinity.
  • Focus on rational forms and integrating the reciprocal function form.

Algebraic Manipulation Technique

  1. Identifying the Highest Power:
    • Locate the highest power of x in the denominator.
  2. Multiplying by Reciprocal Function:
    • Multiply both numerator and denominator by 1 over the highest power.
    • Example: Multiply by 1/x^2 if x^2 is the highest power.
  3. Simplification:
    • Simplify each term by dividing by the highest power.
    • Aim to achieve terms in the form 1/x^n.

Example 1

  • Function with x^2 in Denominator:
    • Multiply by 1/x^2 across both numerator and denominator.
    • Simplify to get terms like 5/x and constants.

Example 2

  • Function with x^4 in Denominator:
    • Multiply by 1/x^4 across both numerator and denominator.
    • Results in terms 3/x^2, 4/x, and constants.

Conclusion

  • The technique is essential for finding limits at infinity.
  • Understanding reciprocal forms within rational functions is key for analysis.

Upcoming Lecture

  • Focus will be on limits at infinity using reciprocal function forms.