CALCULUS

• Limit


  The value that a function's output approaches as the input gets closer to a particular point, but does not necessarily equal that value at the point
• Evaluating the limit of a function


  1. Using the graph

  2. Using tables

  3. Using algebraic approach
• Two-sided limit


  The limit of f(x) as x approaches a from both sides, if the values of f(x) get closer and closer to a real number L
• One-sided limit (left-hand)


  The limit of f(x) as x approaches a from the left, if the values of f(x) get closer and closer to K
• One-sided limit (right-hand)


  The limit of f(x) as x approaches a from the right, if the values of f(x) get closer and closer to K
• Existence of a limit


  lim x→a f(x) = L if and only if lim x→a- f(x) = L = lim x→a+ f(x). Otherwise, the limit does not exist.
• Limits of functions


  • f(x) = x + 2
  • f(x) = x^2 - 4/(x - 2)
  • f(x) = (x - 2)/(x + 2) if x ≠ 2, 0 if x = 2
• Limits of polynomial functions


  • lim x→a f(x) = f(a)
• Limits of polynomial functions


  • lim x→2 5 = 5
  • lim x→3 5 = 5
  • lim x→2 x = 2
  • lim x→3 x = 3
  • lim x→2 (2x^3 + 4x - 1) = 23
  • lim x→3 (2x^3 + 4x - 1) = 65
• Limits of piecewise functions


  • lim x→0 f(x)
  • lim x→1 f(x)
  • lim x→4 f(x)
  • lim x→5 f(x)
  • lim x→-1 f(x)

  
• Properties of limits


  • lim x→a [f(x) + g(x)] = lim x→a f(x) + lim x→a g(x)
  • lim x→a [f(x) - g(x)] = lim x→a f(x) - lim x→a g(x)
  • lim x→a cf(x) = c lim x→a f(x)
  • lim x→a [f(x)g(x)] = [lim x→a f(x)][lim x→a g(x)]
  • lim x→a [f(x)/g(x)] = [lim x→a f(x)]/[lim x→a g(x)] (g(a) ≠ 0)
  • lim x→a √^n f(x) = √^n [lim x→a f(x)] (L > 0 if n is even)
• Evaluating limits using properties

  • lim x→1 [(x - 3)(x^2 - 2)/(x^2 + 1)]
  • lim x→2 [√(2x + 5)/(1 - 3x)]
  • lim x→1 [(1 - 5x)/(1 + 3x^2 + 4x^4)^4]
• Limit of a rational function

  lim x→a r(x) = r(a) for any constant a (h(a) ≠ 0), where r(x) = g(x)/h(x) and g, h are polynomials
• Evaluating limits of rational functions


  • lim x→π (√x - 3√x + 1 - 2x^2)
  • lim x→1 (2x^2 - 4√x + 15)^2
  • lim x→4 (3√(x^2 - 3x + 4)/(2x^2 - x - 1))
  • lim x→-1 (√(3√(x^2/(1 + 2x^2))) - 3)/(x - 3)
• Special cases of limits of quotients


  • lim x→3 (x^2 - 4x + 3)/(x^2 - 7x + 12)
  • lim x→-3 (x - 2)/(x + 3)
• Indeterminate form


  If lim x→a f(x) = 0 and lim x→a g(x) = 0, then lim x→a f(x)/g(x) is indeterminate. The limit may or may not exist.
• Evaluating indeterminate forms


  • lim x→-1 (x^2 + 2x + 1)/(x + 1)
• Rational function


  A function that can be expressed as the ratio of two polynomials (g and h)
• lim x→a r(x) = r(a) for any constant a (h(a) ≠ 0)

  
• Evaluating the limit of a function


  1. lim x→π (√x - 3√x + 1 - 2x²)

  2. lim x→1 (2x² - 4√x + 15)²

  3. lim x→4 (x² - 3x + 4) / (2x² - x - 1)

  4. lim x→-1 (√(x²/3) + 2x²) / (x - 3)
• Indeterminate form


  If lim x→a f(x) = 0 and lim x→a g(x) = 0, then lim x→a f(x)/g(x) is said to be indeterminate
• If lim x→a f(x) = L ≠ 0 and lim x→a g(x) = 0, then lim x→a f(x)/g(x) does not exist
• Evaluating the following limits for the function f(x) = (x+2)²/(x²-4)


  1. lim x→-2 f(x)

  2. lim x→2 f(x)

  3. lim x→0 f(x)
• Evaluating the following limits for the function f(x) = (2x²-3x-2)/(x²+x-6)


  1. lim x→2 f(x)

  2. lim x→1 f(x)

  3. lim x→0 f(x)
• Infinite limit


  lim x→a f(x) = ±∞ if the value of f(x) increases or decreases without bound as x approaches a
• lim x→0 1/x² does not exist
• lim x→0+ 1/x² = ∞
• lim x→0- 1/x² = -∞
• Theorem: Infinite Limits

  • If lim x→a f(x) = c ≠ 0 and lim x→a g(x) = 0, then:

  1. If c > 0 and g(x) → 0 through positive values, lim x→a f(x)/g(x) = ∞

  2. If c > 0 and g(x) → 0 through negative values, lim x→a f(x)/g(x) = -∞

  3. If c < 0 and g(x) → 0 through positive values, lim x→a f(x)/g(x) = -∞

  4. If c < 0 and g(x) → 0 through negative values, lim x→a f(x)/g(x) = ∞
• Vertical asymptote

  The vertical line x = a is a vertical asymptote for the graph y = f(x) if lim x→a+ f(x) = ±∞ or lim x→a- f(x) = ±∞

  
• If f(x) = n(x)/d(x) is a rational function, d(c) = 0, and n(c) ≠ 0, then x = c is a vertical asymptote for the graph of f(x)

  
• Continuity


  A property of functions where the function has no abrupt changes or jumps
• Left Continuous at a Point


  • f(c) exists

  2. lim x→c- f(x) exists

  3. lim x→c- f(x) = f(c)
• Right Continuous at a Point


  • f(c) exists

  2. lim x→c+ f(x) exists

  3. lim x→c+ f(x) = f(c)
• Continuous at a Point

  • lim x→a f(x) exists (it is a real number)

  2. f is defined at x = a

  3. lim x→a f(x) = f(a)

  
• If f satisfies condition 1 but does not satisfy any of the other conditions, then we say that f has a REMOVABLE discontinuity at x = a
• If f does not satisfy condition 1, then we say that f has ESSENTIAL discontinuity at x = a
• Function g(x)


  • g(x) = (x^2 - x - 2)/(x - 2) if x ≠ 2, 0 if x = 2

  
• Function g(x)


  • g(x) = (x^2 - x - 2)/(x - 2) if x ≠ 2, 3 if x = 2

  
• Determine if the function f(x) is continuous at x = 1 and x = 2


  f(x) = 4x - 3 if x < 1, x - 2 if x ≥ 1