Limit Laws

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• Questions?
  • Last problem on problem set? ( lim_x→-2f(x)/x² )
    • Limit laws problem—laws useful for isolating terms you know a limit of in bigger expressions
    • First way of solving
      • Know lim_x→-2( f(x)/x² ) = 1
        • = lim_x→-2f(x) / lim_x→-2(x²) by quotient rule
        • = lim_x→-2f(x) / 4 by evaluating lim_x→-2(x²)
        • i.e., lim_x→-2f(x) / 4 = 1
        • So lim_x→-2f(x) = 4
      • Figure out lim_x→-2f(x)/x
        • = lim_x→-2f(x) / lim_x→-2x
        • = 4 / lim_x→-2x
        • = 4 / -2
        • = -2
    • Another way to solve this
      • Could you make f(x)/x² from f(x)/x
      • lim_x→-2( f(x)/x )
      • = lim_x→-2( (x/x) (f(x)/x) )
      • = lim_x→-2( x f(x)/x² )
      • = lim_x→-2( x ) lim_x→-2(f(x)/x²)
      • = lim_x→-2( x ) × 1
      • = -2 × 1 = -2
• Limit Laws
  • “Limit Laws” part of section 2.2
    • Simplify limits involving arithmetic by taking limits of operands and doing arithmetic on results
    • Sandwich Theorem
      
  • Examples
    • if lim_x→0f(x) = 3 and lim_x→0g(x) = 16, what is
      • lim_x→0( f(x) + g(x) )?
        • Sum rule: lim_x→0( f(x) + g(x) ) = 3 + 16 = 19
      • lim_x→0( f(x) √g(x) )?
        • Product rule & roots rule
        • lim_x→0( f(x) √g(x) )
        • = lim_x→0( f(x) ) lim_x→0√g(x)
        • = 3 × √lim_x→0g(x)
        • = 3 × √16
        • = 12
• Definition of Limit
  
  • lim_x→cf(x) = L if and only if for any ε there is an interval of x values around c that keeps | f(x) - L | < ε
    • This has about half the definition stated formally. Good set-up for…
• Next
  • Formal definition of limit
  • Section 2.3 (“The Precise Definition of a Limit”)

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