Lecture on Limits and Functions

Jul 21, 2024

Lecture on Limits and Functions

Key Topics Covered

  • Basics of Limits
  • Evaluating Limits: Constant, Linear Functions
  • Properties of Limits
  • Advanced Limit Evaluations
  • Limits involving Trigonometric Functions
  • Piecewise Limits
  • Squeeze Theorem

Basics of Limits

Limits of Constants

  • Limit as x approaches a constant (C):
    • Graph: Horizontal line
    • Conclusion: ( \lim_{{x \to a}} C = C )

Limits of Linear Functions

  • Limit as x approaches a variable (X):
    • Graph: Diagonal line passing through the origin
    • Conclusion: ( \lim_{{x \to a}} x = a )

Key Concepts

  • Limit Evaluation: Evaluate the function at a point as it approaches the point from both sides
  • Graphical Interpretation: Understanding the function behavior at specific points on the graph
  • Piecewise Functions: Working through limits of piecewise functions to evaluate at points where function segments connect

Properties of Limits

Key Properties

  1. Addition/Subtraction: ( \lim_{{x \to a}} [f(x) \pm g(x)] = \lim_{{x \to a}} f(x) \pm \lim_{{x \to a}} g(x) )
  2. Multiplication: ( \lim_{{x \to a}} [f(x) \cdot g(x)] = \lim_{{x \to a}} f(x) \cdot \lim_{{x \to a}} g(x) )
  3. Division: ( \lim_{{x \to a}} \left( \frac{f(x)}{g(x)} \right) = \frac{\lim_{{x \to a}} f(x)}{\lim_{{x \to a}} g(x)} ) (provided ( \lim_{{x \to a}} g(x) \neq 0 ))
  4. Power: ( \lim_{{x \to a}} [f(x)]^n = (\lim_{{x \to a}} f(x))^n )

Interpretation

  • Handling Zero Division: When function divisions result in a denominator of zero, careful analysis is required

Advanced Limit Evaluations

Example Problems

  • Evaluation and simplification of complex polynomial limits
  • Understanding limits in rational, polynomial, and higher-degree polynomial functions
  • Handling functions that result in zero crossing out (holes vs. vertical asymptotes)

Steps to Approach

  1. Simplify the equations whenever possible (factoring, rationalizing).
  2. Apply properties from the basics and properties rules to decompose the limit.
  3. Substitute points once functions are simplified.

Limits Involving Trigonometric Functions

Fundamental Trig Limits

  1. Sin/cos Continuous Functions: These functions are continuous everywhere.
  2. Special Limits: Known limits for basic trig functions that can be used as building blocks:
    • ( \lim_{{x \to 0}} \frac{sin(x)}{x} = 1 )
    • ( \lim_{{x \to 0}} \frac{1 - cos(x)}{x} = 0 )
    • ( \lim_{{x \to 0}} \frac{tan(x)}{x} = 1 )

Applying Squeeze Theorem

  • Bounding Functions: Confine the practical limits between known limits for trig functions, apply squeeze theorem to find unknown limits

Strategies

  • Manipulate equations: Often involve multiplying by conjugates or common factors
  • Utilize Pythagorean identities to combine/simplify trig functions

Piecewise Limits

Process Overview

  1. Understand Piecewise Definitions: Evaluate limits for each piece individually at critical points where the function definition changes.
  2. One-Sided Limits: Evaluate left-hand and right-hand limits separately to determine the overall limit

Key Approach

  • Graphical Representation: Visualize where the piecewise functions split and analyze the behavior around these points
  • Compare Limits: One-sided limits should coincide for the overall limit to exist.

Examples and General Problem Solving

Common Problem Types

  • Breaking complex functions into simpler polynomials/higher-degree polynomials
  • Recognize when to apply special limit rules particularly with trig functions
  • Using algebraic manipulation to match the standards, then evaluating limits

Practice Problems

  • Limits involving rationalizing, often involving roots in the numerator or denominator
  • Using limits to evaluate undefined points by simplifying expressions first
  • Verifying the existence of limits via the behavior of functions graphically as well as algebraically

Conclusion

  • Integration of graphical, algebraic, and analytical methods to find limits
  • Deep understanding of limit properties and precise application in varied functions: polynomials, piecewise, trigonometric
  • Realize importance and utility of limits in broader scope mathematics, preparing for applications in calculus and beyond