Lecture Notes: Limits and Graphs

Overview

• Discussion on the behavior of a function f around specific points x = 1, x = 0, and x = 2.
• Introduction to the concept of limits and one-sided limits.
• Explanation of discontinuities and their representation on graphs.

Key Concepts

Limit of a Function

• f of x approaches a certain value as x approaches a specific point.
• Notation: ( \lim_{{x \to a}} f(x) )
• Behavior close to a point can be different from the actual value at that point._

Example: Limit at x = 1

• f(1) = 1, yet as x approaches 1, f(x) ≈ 2.
• ( \lim_{{x \to 1}} f(x) = 2 ) despite ( f(1) = 1 ).
• Discontinuity at (1, 2) if ( f(1) \neq 2 )._

Understanding Continuity

• A function is continuous at a point if the value and the limit coincide.
• Basis for mathematical definition of continuity.

One-Sided Limits

Example: Limit at x = 0

• f(0) = 2
• As x approaches 0 from the left, ( f(x) \approx 2 ).
• As x approaches 0 from the right, ( f(x) \approx 1 ).
• Limit does not exist as ( \lim_{{x \to 0^-}} f(x) \neq \lim_{{x \to 0^+}} f(x) ).

Vertical Asymptotes

Example: Limit at x = 2

• f(2) is undefined; x = 2 is a vertical asymptote.
• As x approaches 2 from the left, ( f(x) \to \infty ).
• As x approaches 2 from the right, ( f(x) \to -\infty ).
• One-sided limits do not exist; thus, the limit as ( x \to 2 ) does not exist.

Exercise Examples

Example 1

• f(1) = 1 because point (1, 1) is on the graph.
• Left-sided limit as ( x \to 1^- ), f(x) approaches 2.
• Right-sided limit as ( x \to 1^+ ), f(x) approaches 0.
• Two-sided limit does not exist as one-sided limits do not agree.

Example 2

• f(2) = 0 because point (2, 0) is on the graph.
• Left-sided limit as ( x \to 2^- ), f(x) approaches 0.
• Right-sided limit as ( x \to 2^+ ), f(x) approaches -\infty.
• Two-sided limit does not exist.

Sketching Graphs with Given Conditions

Example Exercise

• Conditioned on values and limits at various points.
• Important conditions:
  • f(-1) is undefined, right limit as x approaches -1 is 2.
  • f(0) = 1, limit at x = 0 is 0.
  • Vertical asymptote at x = 1.
  • f(2) = 2, left limit at x = 2 is 2.
  • Right limit at x = 2 is 0.
  • Limit as x approaches 3 from the left is 1.
• Illustration of possible graph configurations.

Conclusion

• Understanding limits and behavior near points enhances comprehension of function continuity and discontinuity.
• Graphs represent mathematical behavior and serve as visual aids in limit concepts.