Lecture Notes: Limits at Infinity

Key Concepts

• Limits at Infinity: Understanding how a function behaves as x approaches positive or negative infinity.
• Asymptotes: When the limit of a function as x approaches a number goes to infinity, it indicates the presence of an asymptote.
• Removable Discontinuity (Hole): Occurs when both the numerator and denominator equal zero at the same point, allowing simplification.
• Vertical Asymptote: Occurs when the denominator equals zero but cannot be canceled out.

Finding Discontinuities

• Discontinuities are found where the denominator of a rational function equals zero.
• Types of Discontinuities:
  • Hole: When you can cancel the factor out.
  • Asymptote: When you cannot cancel the factor out.

Determining Limits and Asymptotes

• Sign Analysis Test: Used to determine behavior around vertical asymptotes by analyzing the sign of the function in intervals.
• Horizontal Asymptotes: Determined by the behavior of the function as x approaches infinity or negative infinity.

Calculating Limits at Infinity

• Key Formula: For a rational function ( \frac{1}{x^n} ), as ( x \rightarrow \infty ), the function approaches 0.
• Polynomials: Limits of polynomials as ( x \rightarrow \infty ) follow the behavior of the leading term.
• Limit Rules: Application of limit rules for functions as ( x \rightarrow \pm \infty ) remain consistent.

Examples

• Rational Functions: Simplifying limits by dividing by the highest power of x in the denominator.
• Use of Exponents: Understanding the impact of dividing by the largest power in the denominator, and the effect on the limit.
• Handling Complex Functions: Utilize conjugates and rationalizing techniques when functions involve roots.

Additional Insights

• Behavior of Polynomials: Polynomials either go to positive or negative infinity based on the highest power term.
• Cube Root Example: Demonstrates using the cube root to simplify the limit problem.
• Piecewise Definition of Absolute Value: Important for evaluating limits approaching negative infinity.
• Undefined Limits: Be aware of limits that result in undefined conditions such as square roots of negative numbers.

Important Concepts to Remember

• Consider Both Positive and Negative Infinity: Horizontal asymptotes can differ when approaching from positive or negative infinity.
• Absolute Value Considerations: Necessary when dealing with limits as x approaches negative infinity.
• Critical Thinking in Limits: Approach limits by analyzing the behavior and operations on the function, rather than straightforward calculations.