Lecture Notes on Limits

Brief Overview of Limits

• Discussion on limits as a function where:
  • y = f(x)
  • x is the independent variable and y is the dependent variable.
• Example: If f(x) = 3x + 5, then:
  • When x = 1, y = 8.
  • As x approaches 1, y approaches 8.

Limit Notation

• Written as: lim (x → a) f(x) = L
  • L must be both definite and finite.
  • Importance of L being definite and finite.

Right-Hand Limit (RHL) and Left-Hand Limit (LHL)

• Right-Hand Limit (RHL):
  • Denoted as lim (x → a⁺) f(x) (x approaching from the right).
• Left-Hand Limit (LHL):
  • Denoted as lim (x → a⁻) f(x) (x approaching from the left).
• If RHL ≠ LHL, then the limit does not exist.

Example: Greatest Integer Function Graph

• The graph demonstrates how RHL and LHL can differ at points where the function is not continuous.
• Notably, at x = 2:
  • RHL = 2 (approaching from right)
  • LHL = 1 (approaching from left)
  • Therefore, lim (x → 2) greatest integer(x) does not exist.

Key Takeaways on Limits

• Existence of Limit vs. Function:
  • A function can exist at a point where the limit does not exist.
  • Example: The function exists at x = 2, but the limit does not.

Methods of Evaluating Limits

1. Direct Substitution Method:

   • Substitute the value of x directly into the function.
   • Acceptable when not resulting in an indeterminate form (0/0 or ∞/∞).
   • Example: lim (x → 1) (3x² + 4x + 5) results in 12.

2. Factorization Method:

   • Used when direct substitution yields an indeterminate form.
   • Factor polynomials and cancel common factors.
   • Example: lim (x → 2) (x³ - 1)/(x - 1) can be factored and simplified to find the limit.

3. Rationalization Method:

   • Necessary when square roots are involved that cause indeterminate forms.
   • Multiply by the conjugate to simplify.
   • Example: lim (x → 0) (√(2 + x) - √2)/x can be rationalized to solve the limit.

Practical Examples

• Example of Direct Substitution:
  • lim (x → 2) (3x² + 4x + 5) = 12.
• Example of Factorization:
  • Find lim (x → 2) (x² - 4)/(x - 2), leading to a limit of 2 after canceling the common factor.
• Example of Rationalization:
  • Evaluate lim (x → 0) (√(2 + x) - √2)/x to find the limit after rationalizing.

Summary of Methods

• Always check for direct substitution first.
• If indeterminate, consider factorization or rationalization.
• Understanding how to manipulate functions is crucial for limit evaluation.

Next Steps

• Upcoming topics will cover standard limits and infinite limits.