Lecture on Limits

Introduction to Limits

• Understanding limits and how to evaluate them both analytically and graphically.
• Example function: f(x) = (x^2 - 4) / (x - 2).
• Direct substitution leads to an indeterminate form 0/0.

Evaluating Limits Analytically

Direct Substitution

• Example: Plug in values close to the target.
  • f(2.1) ≈ 4.1, f(2.01) ≈ 4.01.
  • Limit as x approaches 2 is 4.

Factoring Technique

• Factor expressions to simplify.
• x^2 - 4 = (x + 2)(x - 2).
• Cancel the term x - 2 and use direct substitution.
• Resulting limit is 4.

Polynomial Limit

• Example: Limit as x approaches 5 of x^2 + 2x - 4.
• Use direct substitution.
  • 5^2 + 2(5) - 4 = 31.

Factoring Cubes

• Example: Limit as x approaches 3 of (x^3 - 27) / (x - 3).
• Utilize the difference of cubes formula.
• a^3 - b^3 = (a - b)(a^2 + ab + b^2).
• Limit is 27 after canceling and substitution.

Complex Fractions

• Example: Limit as x approaches 3 of (1/x - 1/3) / (x - 3).
• Multiply by LCD 3x, simplify.
• Cancel terms, use direct substitution.
  • Final limit is -1/9.

Limits Involving Square Roots

• Example: Limit as x approaches 9 of (sqrt(x) - 3) / (x - 9).
• Multiply by conjugate (sqrt(x) + 3).
• Simplify, cancel terms, substitute.
  • Limit is 1/6.

Complex Fractions with Radicals

• Example: Limit as x approaches 4 of (1/sqrt(x) - 1/2) / (x - 4).
• Use common denominator and conjugate.
• Factor, simplify, substitute.
  • Limit is -1/16.

Evaluating Limits Graphically

Understanding Graphs

• Evaluate limits from left and right sides.
• If left and right limits don't match, limit doesn't exist.

Examples

• x approaches -3:

  • Left: Limit is 1.
  • Right: Limit is -3.
  • Overall limit does not exist.

• x approaches -2:

  • Limit exists at -2 (both sides match).

• x approaches 1:

  • Left: 1, Right: 3.
  • Overall limit does not exist.
  • Function value: 2.

• x approaches 3:

  • Vertical asymptote, limit does not exist.
  • Function value is undefined.

Types of Discontinuities

• Jump discontinuity: Graph has a gap; non-removable.
• Hole: Removable discontinuity.
• Infinite discontinuity: Vertical asymptote; non-removable.