Lecture Notes: Greatest Integer Function (Floor Function)

Introduction

• Definition: The greatest integer function, also known as the floor function, maps a real number to the largest integer less than or equal to it.
• Notation: ( \lfloor x \rfloor ) represents the greatest integer of ( x ).

Evaluating the Greatest Integer Function

• Example 1: Greatest integer of 0.7
  • Process: Identify integers less than 0.7. The greatest is 0.
  • Visualization: Use a number line; 0.7 falls between 0 and 1.
  • Result: ( \lfloor 0.7 \rfloor = 0 )
• Example 2: Greatest integer of 1.8
  • Result: ( \lfloor 1.8 \rfloor = 1 )
• Example 3: Greatest integer of 2.3
  • Result: ( \lfloor 2.3 \rfloor = 2 )

Negative Values

• Example 4: Greatest integer of -1.6
  • Result: ( \lfloor -1.6 \rfloor = -2 )
• Example 5: Greatest integer of -2.8
  • Result: ( \lfloor -2.8 \rfloor = -3 )

Graphing the Greatest Integer Function

• Steps:
  • Plot integer points on a number line or graph.
  • Horizontal segments between integers.
  • Closed circle at the beginning of a segment, open circle at the end.
• Example:
  • ( \lfloor x = 0.3 \rfloor ) and ( \lfloor x = 0.6 \rfloor ) both equal 0. Segment from 0 to 1 on graph.

Practicing With Positive and Negative Values

• Exercise:
  • ( \lfloor 3.2 \rfloor = 3 )
  • ( \lfloor 7.8 \rfloor = 7 )
  • ( \lfloor -3.2 \rfloor = -4 )
  • ( \lfloor -4.7 \rfloor = -5 )
• Integer Values:
  • ( \lfloor 4 \rfloor = 4 )
  • ( \lfloor -5 \rfloor = -5 )

Evaluating Limits with Greatest Integer Functions

• Concept: Evaluate one-sided limits when the function involves the greatest integer.
• Example 6: Limit as ( x \to 0 ) of ( \lfloor x \rfloor )
  • Left-sided limit: Approximate with negative numbers, e.g., -0.1. Result: -1
  • Right-sided limit: Approximate with positive numbers, e.g., 0.1. Result: 0
  • Conclusion: Limit does not exist since left and right limits do not match.

Complex Limit Problems

• Example 7: Limit as ( x \to 2^- ) of ( \lfloor x \rfloor + 3x )

  • Substitute values just less than 2, e.g., 1.9
  • ( \lfloor 1.9 \rfloor = 1 )
  • Calculate: ( 1 + 3(2) = 7 )

• Example 8: Limit as ( x \to -3^- ) of ( 5 - 2\lfloor x \rfloor )

  • Substitute values like -3.1
  • ( \lfloor -3.1 \rfloor = -4 )
  • Calculate: ( 5 - 2(-4) = 13 )

• Example 9: Limit as ( x \to 3 ) of ( 2 - \lfloor -x \rfloor )

  • Check both sides: Left uses values like 2.9, right uses 3.1.
  • Left side result: 5, Right side result: 6
  • Conclusion: Limit does not exist, sides differ.

• Example 10: Limit as ( x \to -2^- ) of ( \lfloor -x \rfloor )

  • Substitute values like -2.1
  • ( \lfloor 2.1 \rfloor = 2 )
  • Conclusion: Result is 2.