The Skys The Limit

1
The Skys The Limit

• AP Calculus Review
• Of
• Limits

2
What is a Limit?

• The height that a function intends to reach at a
  given x-value, whether or not it actually reaches
  it, is a limit.
• A limit can exist even if there is a hole in the
  graph.

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When Does a Limit Exist?

• The left-hand limit must exist at xc.
• The right-hand limit must exist at xc.
• The left- and right-hand limits at c must equal.

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When Does A Limit Not Exist?

• If it has differing behavior from the left and
  right side, the limit does not exist.
• If it increases or decreases infinitely
  (unbounded behavior), the limit does not exist.
• If it oscillates infinitely, the limit does not
  exist.

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How Are Limits Evaluated?

• Substitution Method
• Factoring/Cancellation Method
• Conjugate Method
• Common Denominator Method

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Practice Problems
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Answers

• 14
• 5/3
• 13
• Limit

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Limits and Infinity

• If the substitution method results in any nonzero
  number divided by zero, then the function has
  unbounded behavior, which indicates that the
  function has a vertical asymptote.
• If the substitution method results in zero
  divided by zero, then the function has a hole in
  it.

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Vertical Asymptotes

• To find where vertical asymptotes occur,
  determine where the functions denominator equals
  zero by factoring the numerator and denominator.
• Remember any expressions that cancel out
  represent a hole, not an asymptote.

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Horizontal Asymptotes

• Horizontal asymptotes occur if the function has a
  limit as x approaches positive or negative
  infinity.
• It represents the functions intended height as x
  gets larger or smaller.

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Evaluate Limits at Infinity

• Use BOB, SAME, and TUB rules for rational
  expressions
• Special Limit Theorems

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Continunity

• A finction f is continuous at c if the following
  three conditions exist
• f ( c ) is defined.
• Lim f (x) exists.
• x-gtc
• Lim f (x) f ( c )
• x-gtc