Formal Definition of Limit, Part 2

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• Questions?
• Formal Definition of Limit
  • If 0 < | x - c | < δ then | f(x) - L | < ε
  • A Strategy
    • Start with | f(x) - L | < ε
    • Try to rewrite into | x - c | < δ
  • Prove that lim_x→14x = 4 by finding a general formula for finding δ given ε
    • |4x - 4| < ε
    • 4|x - 1| < ε
    • |x - 1| < ε/4 = δ
    • Now should you show | x - 1 | < ε/4 implies |4x - 4| < ε?
      • Technically that is what the definition of “limit” calls for
      • But in this case (and many others) the algebra you did to rewrite the inequality involving ε into one defining δ is reversible, and so you needn’t explicitly show that | x - 1 | < ε/4 implies |4x - 4| < ε
  • Prove that lim_x→0x³ = 0
    • | x³ - 0 | < ε
    • | x³| < ε
    • -ε < x³ < ε
    • (-ε)^1/3 < x < ε^1/3
    • -(ε^1/3) < x < ε^1/3 (because cube root is odd)
      • (“Odd function” f(x) is one where f(-x) = -f(x))
    • |x| < ε^1/3
    • |x - 0| < ε^1/3
• 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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