Math 1350: Survey of Calculus - Limits Video B

Overview

• Instructor: Mark Barsamian, Ohio University
• Course: Math 1350 (Survey of Calculus), second video installment
• Topic: Limits from an analytic perspective (functions given by formulas)

Key Concepts

• Limits: Concept revisited from a formulaic perspective
• Analytical Approach: Focus on functions presented with formulas rather than graphs
• Estimating Limits: Use of tables to estimate limit values
• Properties of Limits: Introduction to theorems and analytical techniques

Definitions and Concepts

• Limit Notation: ( \lim_{{x \to c}} f(x) = L )
  • x: Variable approaching a number c
  • f(x): Function
  • L: Limit value as x approaches c

Example 1: Polynomial Function

• Function: ( f(x) = -7x^2 + 13x - 25 )
• Part A: Find ( f(-2) = -79 )
• Part B: Use table to estimate ( \lim_{{x \to -2}} f(x) = -79 )
  • Table with x-values approaching -2 from both sides
  • Conclusion: ( f(x) ) values trend towards -79
• Remarks: Limit estimation vs calculating exact y-value, potential discrepancies

Theorems on Limits

1. Properties of Limits
   • Sum of Limits: ( \lim(f+g) = \lim f + \lim g )
   • Product of Limits with constant: ( \lim(kf) = k \cdot \lim f )
   • Constant Function: ( \lim k = k )
   • Identity Function: ( \lim x = c )
2. Limits of Polynomial Functions
   • ( \lim_{{x \to c}} f(x) = f(c) ) for polynomial functions
3. Limits of Rational Functions
   • If denominator is non-zero at ( x = c ): ( \lim_{{x \to c}} r(x) = r(c) )
4. Limits of Quotients
   • If numerator limit is non-zero and denominator limit is zero, then limit does not exist

Example 2: Radical Function

• Function: ( f(x) = \sqrt{24 + x^2} )
• Part A: Find ( f(5) = 7 )
• Part B: Find ( \lim_{{x \to 5}} f(x) = 7 )
  • Use theorem 2.8 to manipulate limit expression inside radical

Example 3: Rational Function

• Function: ( f(x) = \frac{(x-1)(x-3)}{(x-2)(x-4)} )
• Part A: Find ( f(1) = 0 )
• Part B: Find ( \lim_{{x \to 1}} f(x) = 0 )
  • Use theorem for rational functions with non-zero denominator
  • ( x = 1 ) does not cause denominator to be zero
• Part C: ( f(3) ) does not exist (denominator = 0)
• Part D: ( \lim_{{x \to 3}} f(x) ) does not exist
  • Limits of numerator = -4 (non-zero)
  • Limits of denominator = 0

Conclusion

• Use of analytical methods and theorems simplifies limits calculation
• Theorems help avoid estimation and provide precise limit values
• Further videos will continue exploring limits and calculus topics