Formal Definition of Limit

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• Questions?
• Formal Definition of Limit
  • Section 2.3
    • Variables mean...?
      • ε (epsilon)
      • δ (delta)
      • L
      • c
        
    • Absolute values...?
      • Distance between f(x) and L: f(x) - L
      • Distance between x and c: x - c
      • Absolute value lets you ignore which is bigger
  • Values for ε, δ?
    • You might be given a specific ε, for example suppose you believe that lim_x→14x = 4, and I challenge you with ε = 0.5. Find a δ that supports your belief
      • Short proof:
        • 3.5 < 4x < 4.5
        • 3.5 < 4x; 4x < 4.5
        • 7/8 < x; x < 9/8
      • Book version of above
        • | f(x) - L | < ε
        • | 4x - 4 | < 0.5
        • -0.5 < 4x - 4 < 0.5
        • 3.5 < 4x < 4.5
      • δ is distance from c = 1
        • δ = 1/8
        • Or, done the textbook way...
          • 7/8 < x < 9/8
          • -1/8 < x - 1 < 1/8
          • | x - 1 | < 1/8
      • Plot this with R
        

        > x = seq( 0, 2, by=0.02 )
        > y = 4 * x
        > plot( x, y, type="l" )
        > lines( c(0, 2), c(3.5, 3.5), col="red")
        > lines( c(0,2), c(4.5,4.5), col="red")
        > lines( c(7/8,7/8), c(0,5),col="blue")
        > lines( c(9/8,9/8), c(0,5),col="blue")

        
        
    • Or you might need to treat ε as a variable in order to prove a limit equation, e.g., prove that lim_x→14x = 4 by finding a general formula for finding δ given ε
      • First try...
        • | 4x - 4 | < ε
        • -ε < 4x - 4 < ε
        • 4-ε < 4x < 4+ε
        • (4-ε)/4 < x < (4+ε)/4
        • And we could keep going this way and get to the answer. But...
      • Can we more directly get something of the form 0 < | x - 1 | < δ
        • | 4x - 4 | < ε
        • 4 |x-1| < ε
        • |x-1| < ε/4 = δ
• Next
  • Limits miscellany
  • “Treasure hunt” (for chocolate) in sections 2.4 - 2.6
    • What, precisely, does it take to make a function continuous at x = c? On an interval?
    • What does the Intermediate Value Theorem say?
    • What does it mean to talk about lim_x→±∞f(x)?
    • What does it mean to say that lim_x→cf(x) = ±∞?
    • What is a 1-sided limit?
    • What is an asymptote?

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