Lecture on Finding Limits at Infinity

Key Concepts

• Limit at Infinity: Understanding how a function behaves as x approaches infinity or negative infinity.
• Polynomial Functions: Focus on the term with the highest degree as x approaches infinity.

Examples and Explanations

Limits of Polynomial Functions

• Function: x²

  • Limit as x approaches infinity: x² grows larger, so it approaches infinity.
  • Limit as x approaches negative infinity: (-x)² = x², still positive, so it approaches infinity.

• Function: x³

  • Limit as x approaches infinity: x³ grows larger, so it approaches infinity.
  • Limit as x approaches negative infinity: (-x)³ = -x³, so it approaches negative infinity.

Limits of Polynomial Expressions

• Expression: 5 + 2x - x³

  • As x approaches negative infinity: Simplify to -x³ (ignoring smaller terms like 5 and 2x).
  • Result: Positive infinity because -(-∞)³ results in positive infinity.

• Expression: 3x³ - 5x⁴

  • As x approaches negative infinity: Focus on -5x⁴, which dominates.
  • Result: Negative infinity because -5 times (-∞)⁴ is negative.

Limits of Rational Functions

• Function: 1/x

  • As x approaches infinity: 1/x approaches 0 since the fraction gets smaller.
  • General rule: If a rational function is bottom-heavy (denominator has higher degree), limit is 0.

• Function: (5x + 2) / (7x - x²)

  • Bottom-heavy: Degree of denominator (2) > numerator (1).
  • Limit: 0

Rational Functions with Equal Degree

• Function: (8x² - 5x) / (4x² + 7)
  • Equal degree: Divide coefficients of terms with highest degree.
  • Result: 8/4 = 2

Rational Functions with Top-Heavy Degree

• Function: (5x - 7x³) / (2x² + 14x³ - 9)

  • Degrees are equal (3); result is coefficient division: -7/14 = -1/2

• Example: (5x + 6x²) / (3x - 8)

  • Top-heavy: Focus on 6x² over 3x.
  • Result: Simplifies to 2x, which approaches infinity as x goes to infinity.

Simplification Techniques

• Remove insignificant terms for quicker calculations.
• Balance top and bottom by multiplying with appropriate power of x to simplify rational functions.

Summary

• For polynomial terms, highest degree terms dominate.
• Rational functions can be analyzed based on degree comparison: bottom-heavy, equal degree, or top-heavy.
• Use simplification to efficiently find limits and understand function behavior at infinity.