Princeton Review AP Calculus BC, Chapter 3: Limits and Continuity

LIMITS

Limit: The value that a function approaches as the variable within the function gets nearer to a particular value.

  • To find the limit of a simple polynomial, plug in the number that the variable is approaching

\ 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

   

  • 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

\ 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

\ CONTINUITY

Discontinuity

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

\ Defining 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

\ Confirming Continuity Over an Interval

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

\ 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

\ 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

\ 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

\ 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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