📊

Understanding and Evaluating Calculus Limits

Jun 5, 2025

View transcript

Calculus Lecture Notes: Understanding Limits

Introduction to Limits

  • Function Example: Given ( f(x) = \frac{x^2 - 4}{x - 2} ), determine ( \lim_{x \to 2} f(x) ).
  • What is a Limit?
    • As ( x ) approaches a value, what does ( f(x) ) approach?_

Techniques to Evaluate Limits

  1. Direct Substitution

    • Substitute ( x = 2 ): ( \frac{2^2 - 4}{2 - 2} = \frac{0}{0} ) (undefined)
    • Try values close to 2:
      • ( f(1.9) \approx 3.9 )
      • ( f(1.99) \approx 3.99 )
    • Conclusion: As ( x \to 2, f(x) \to 4 ).

  2. Algebraic Simplification

    • Factor and simplify: ( x^2 - 4 = (x + 2)(x - 2) )
    • Cancel ( x - 2 ): Evaluate ( \lim_{x \to 2} (x + 2) = 4 )._

Example Problems

Problem 1

  • Limit Problem: ( \lim_{x \to 3} (x^2 + 5x - 4) )
    • Direct substitution: 20_

Problem 2

  • Limit Problem: ( \lim_{x \to 3} \frac{x^2 - 8x + 15}{x - 3} )
    • Factor: ( x^2 - 8x + 15 = (x - 3)(x - 5) )
    • Cancel ( x - 3 ): Evaluate ( \lim_{x \to 3} (x - 5) = -2 ).

Problem 3

  • Limit Problem: ( \lim_{x \to 4} \frac{1/x - 1/4}{x - 4} )
    • Multiply by common denominator ( 4x ): Simplify to find limit as ( -\frac{1}{16} ).
    • Verification using values close to 4._

Problem 4

  • Limit Problem: ( \lim_{x \to 9} \frac{\sqrt{x} - 3}{x - 9} )
    • Multiply by conjugate: Evaluate to find ( \frac{1}{6} )._

Evaluating Limits Graphically

  • One-Sided Limits
    • Left-Side Limit: Approaches from left.
    • Right-Side Limit: Approaches from right.
  • If left and right limits differ, the limit does not exist.

Types of Discontinuities

  1. Jump Discontinuity (Non-Removable): Different left and right limits.
  2. Removable Discontinuity (Hole): Limit exists but differs from function value.
  3. Infinite Discontinuity: Involves vertical asymptotes (limit tends to infinity).
  4. Continuous Function: Limits from both sides and function value are the same.

Practice Questions

  • Identify Limits and Discontinuities:
    • Evaluate the limit as ( x \to -3, -1, -2, 1 ) from left, right, and both sides.
    • Determine function values at these points and identify type of discontinuity.

These notes cover the fundamental concepts and methods for understanding and evaluating limits in calculus, along with different types of discontinuities that can occur in functions. Use them to practice problems and prepare for exams. Each technique and example highlights a different aspect of solving limits, ensuring a comprehensive understanding of the topic.

Full transcript