Lecture Notes on Limit Techniques and Squeeze Theorem

Introduction

• Continuing with section 2.3: discussing additional limit techniques.
• Focus on finding common denominators and using the squeeze theorem.

Common Denominator Technique

• Addition/Subtraction of Fractions:

  • Ensure fractions have the same denominator.
  • Multiply fractions by necessary terms to achieve a common denominator.
  • Formula: ( \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}).
  • Check that denominators are not zero.

• Example Problem:

  • Limit as ( x \to 2 ) of ( \frac{1}{x} - \frac{1}{2} ) divided by ( 2 - x ).
  • Direct substitution results in ( \frac{0}{0} ), an indeterminate form.
  • Combine fractions using a common denominator: ( \frac{2 - x}{2x} ).
  • Simplify, cancel terms, and substitute ( x = 2 ) to find the limit: ( \frac{1}{4} ).

Combined Techniques Example

• Complex Limit Problem:
  • Limit as ( x \to 9 ) of ( x^{-1/2} - \frac{1}{3} ) over ( x - 9 ).
  • Rewrite ( x^{-1/2} ) as ( 1/\sqrt{x} ).
  • Combine fractions with a common denominator, then multiply by a conjugate.
  • Simplify and cancel terms to find the limit.

Piecewise Functions and Limits

• Definition:

  • A function defined in multiple segments.
  • Evaluate limits by checking from both sides (left and right limits).

• Example:

  • Evaluating limits at transition points using piecewise-defined functions.
  • Technique involves checking the limit from both directions.
  • Ensure continuity by setting left limit equal to right limit.

Squeeze Theorem

• Concept:

  • Requires three functions, ( f(x) \leq g(x) \leq h(x) ) near a value ( c ).
  • If both outer functions approach the same limit ( L ), the middle function must also approach ( L ).

• Graphical Representation:

  • A function squeezed between two others as ( x \to c ).
  • Used when direct evaluation is not possible.

• Examples:

  • Squeeze Example 1: ( \lim_{x \to 0} \sqrt{x} \cdot e^{\sin(\pi/x)} = 0 ).
  • Squeeze Example 2: Applying squeeze theorem to evaluate limits of functions with sine/cosine terms.

Additional Concepts

• Strictly Increasing Functions:

  • Exponential functions are strictly increasing if base ( b > 0 ).
  • Inequality properties remain unchanged when exponentiating.

• Developing Inequalities for Squeeze Theorem:

  • Utilize the range of sine and cosine functions.
  • Exponentiate across inequalities to maintain direction.

Conclusion

• Skill Development:
  • Mastery of these techniques allows for solving complex limit problems efficiently.
  • Understanding piecewise functions and the squeeze theorem is crucial for calculus.

---

These notes cover advanced techniques in evaluating limits, focusing on algebraic manipulation for fractions and strategic use of the squeeze theorem in calculus. The use of piecewise functions and exponential functions in these contexts is also explored.