Limits of Multivariable Functions

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• Misc
  • Math department scholarships
    • Open to sophomores and juniors
    • Apply by March 23
    • See email invitation and/or dept. scholarships web page
    • (www.geneseo.edu/math/scholarships)
  • Colloquium
    • “Boundary Value Problems: From Integer to Fraction”
      • Boundary value problem: scale up familiar single-variable initial value problems to more general differential equations on multivariable functions over regions, with initial conditions commonly given at some boundary of region
        
    • Dr. Min Wang (faculty candidate)
    • Friday, March 4, 2:30 PM, Newton 201
    • Usual extra credit for colloquium write-ups
• Exam Review
  • Done asynchronously while I’m at a conference
  • Read and edit Google doc of emerging solutions
• Questions?
• Limits of Multivariable Functions
  • Section 14.2
    • lim f(x,y) = L as (x,y) approaches (x₀,y₀)
      • | f(x,y) - L | < ε whenever 0 < √((x-x₀)2+(y-y₀)2) < δ
      • This means what?
        
    • Function is continuous if limits exist
    • Rules for limits of multivariable functions similar to rules for single-variable functions
    • Examples? (see below)
    • Extreme value theorem?
      • Single variable version: over interval [a,b], continuous f(x) has at least 1 local minimum and 1 local maximum
      • Multivariable version: same holds over closed bounded regions
  • Examples
    • Auditorium function from yesterday
      • z = f(x,y) = √ (r² - (x-y)²/2 ), r constant
      • No continuity issues, non-trivial limits
      • So limits can be found by plugging x, y, into f
    • lim_{(x,y)→(1,0)}( (xy - x - y + 1) / (x-1) )
      • = lim_{(x,y)→(1,0)}( (y(x-1) - (x - 1)) / (x-1) )
      • = lim_{(x,y)→(1,0)}(y - 1)
      • = -1
    • lim_{(x,y)→(0,0)}( 3x³ / (x³+y) )
  • Problem set
• Next
  • (Tuesday)
  • Finish limits
    • Think about lim_{(x,y)→(0,0)}( 3x³ / (x³+y) )
  • Partial derivatives
  • Read section 14.3

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