Limits

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• Misc
  • Math Learning Center open in South 332
    • 10:30 AM - 3:30 PM + 7:00 PM - 10:00 PM most days
    • Exceptions
      • W afternoon close at 2:30 (open evening though)
      • No evening hours Fridays
      • Closed Saturdays
      • No daytime hours Sundays
  • Other editions of textbook
    • Not recommended
    • Section, exercise, and page numbers may trip you up
    • But if you’re willing to take the risks you can
• The Basics of Limits
  • “Limits of Function Values” part of section 2.2
    • Holes in graphs where function value doesn’t follow pattern
    • e.g., because function is undefined there, or because function is defined piecewise
      • Sometimes because of division by zero
      • Fix by cancelling out common factors in numerator and denominator
    • lim_x→cf(x) = L means “limit of f(x) as x approaches c is L”
    • Some functions have no limit at certain points because they are discontinuous there
      
  • Look at this in R
    • e.g., for function f(x) = (x²-x)/x
      

      > x = seq(-2,2,by=0.1)
      > x
       [1] -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1  0.0
      [22]  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9  2.0
      > y = (x^2 - x) / x
      > y
       [1] -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1  NaN
      [22] -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0
      > plot( x, y, type="l")

      
      
    • Note gap in plot where (x²-x)/x is undefined and so R can’t calculate a value
      • Make this gap as small as you like by making step size between x values smaller
        

        x = seq(-2,2,by=0.01)

  • Examples: calculate some limits
    • lim_x→0f(x) for the function graphed here
      
      • Visually, limit is 2 or a little less
    • lim_x→1f(x) for the same function
      • Limit is 1 despite function being undefined there
    • lim_x→0|x|
      • Visualizing graph of |x|, limit must be 0
      • Could also figure that limit = |0| because |x| is continuous at x = 0
      • But note that a function similar to |x| but discontinuous at x = 0 could also have a limit of 0 as x approached 0
        
  • Problem Set
    • Handout
• Next
  • Some rules for calculating limits
  • Read rest of section 2.2
    • Pages 51 - 55

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