Lecture Notes: Understanding Limits

Overview

• Limits are a fundamental concept in calculus.
• Often introduced in high school, with more depth explored at the university level.
• Basic approach involves substituting the limit value into the expression directly.

Basic Method for Evaluating Limits

1. Substitute Directly

   • Plug the value that x approaches directly into the expression.
   • If no errors occur on the calculator, the process is straightforward.
   • Example: For (f(x) = x^2 + 3x + 4), substituting 0 gives 4.

2. Handling Errors

   • Errors often occur when the denominator becomes zero.
   • Always check if there is a denominator issue when substituting.

Complex Cases

• Factoring

  • If direct substitution leads to an error, try factoring.
  • Example:
    • Given (\lim_{x \to -1} \frac{x^2 - 3x - 4}{x + 1}), factor the numerator.
    • Cancel common terms (x + 1), and substitute (-1) into the simplified expression.
    • Result: (-5).

• Zero Numerator

  • If the numerator is zero but the denominator is non-zero, the result is 0.
  • Example: (\frac{0}{2} = 0)._

Steps for Solving Limits

• Direct Substitution
  • If the direct substitution gives an error, proceed to factoring or simplifying.
• Write Limit Notation Until Substitution
  • Keep writing (\lim_{x \to a}) until the substitution is complete.
• Cancel Common Factors
  • Simplify the expression by canceling terms._

Summary

• Key Steps
  • Substitute the value directly.
  • If an error occurs, factorize and simplify.
  • Always aim for no zero in the denominator.
• Examples Recap
  • Successfully managing limits involves substitution, factoring, and simplification.
• Main Takeaway
  • Simplify where possible and ensure substitutions do not result in a zero denominator.