Unit 1: Limits and Continuity

Limits

• Limits are the value that a function approaches as the variable within the function gets nearer to a particular value.
• We don’t really care what’s happening at the point, we care about what’s happening around the point
• To find the limit of a simple polynomial, plug in the number that the variable is approaching

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Ways to Find Limits

• Look on a graph to see what it approaches

  • If the graph approaches two different values for the same number, the limit does not exist

• Estimate from a table

• Algebraic Properties

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• Algebraic Manipulation

  • You can factor the numerator and denominator, then cancel any removable discontinuities
  • This is mostly useful if you get limits where the denominator is equal to 0

• For example, (x+3)(x+2)/(x+3)(x-3)

  • (x+3) is able to be removed → removable discontinuity

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Squeeze Theorem

• Conditions
  • For all values of x in the interval that contains a, g(x) ≤ f(x) ≤ h(x)
  • g and h have the same limit as x approaches a
• lim g(x) = L, lim h(x) = L, therefore lim f(x) = L
• Trig limits as x approaches 0:
  • lim [sin(x)/x] = 1
  • lim [(cos(x)-1)/x] = 0
  • lim [sin(ax)/x] = a
  • lim [sin(ax)/sin(bx)] = a/b

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Continuity

• Jump Discontinuity

  • Occurs when the curve “breaks” at a particular place and starts somewhere else
  • The limits from the left and the right will both exist, but they will not match

• Essential/Infinite Discontinuity

  • The curve has a vertical asymptote

• Removable Discontinuity

  • An otherwise continuous curve has a hole in it
  • “Removable” because one can remove the discontinuity by filling the hole

• Continuity Conditions

  • For f(x) to be continuous when x=c:
  • f(c) exists
  • the limit as x→c exists
  • lim f(x) = f(c)
    • x→c

• A function is continuous on an interval if it is continuous at every point on that interval

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Removing Discontinuities

• You can remove a discontinuity by redefining the function without that point in the domain
• This is frequently done by factoring out a common root between the numerator and denominator

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Limits and Asymptotes

• Vertical asymptote: a line that a function cannot cross because the function is undefined there
• Horizontal asymptote: the end behavior of a function
  • A horizontal asymptote can be crossed

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Horizontal Asymptote Rules

• If the highest power of x in a rational expression is in the numerator, then the limit as x approaches infinity is infinity: there is no horizontal asymptote
• If the highest power of x is in the denominator, then the limit as x approaches infinity is zero and the horizontal asymptote is the line y=0
• If the highest power is the same, then the limit is the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator

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Intermediate Value Theorem

• Guarantees that if a function f(x) is continuous on the interval [a,b] and C is any number between f(a) and f(b), ten there is at least one number in the interval [a,b] such that f(x) = C

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