Formal Definition of a Limit

1
Formal Definition of a Limit

• AP Calculus I
• Ms. Hernandez

2
Formal Definition of a Limit

• The idea of the limit is that as x gets closer to
  a value c, without being equal to c, f(x) must
  get closer to some value L that we call a limit.
  
• Calculus uses ? to represent how far f(x) is from
  L . It is a positive, but very small increment
  which creates an interval about L .
• The ? value is the error we allow in the output.
• The corresponding value for the interval of the x
  values around c is d . This is also a positive,
  but very small value.
• The d values depend on the ? values .
• We use the ? value of how much error is
  acceptable in the output, how close we want to
  come to our target value, L, to find d .

3
Formal Definition

• The limit of f(x) as x approaches c is L
• if and only if, given ?  gt 0, there exists d  gt 0
  such that 0 lt x - c lt  d implies that
  f(x) - L lt  ?.
• Some resources use e for ? and d for d
• These symbols are inter- changeable

4
What does that mean?

• Recall limit definition That the limit of f(x)
  as x approaches c is L if and only if, given ?
   gt 0, there exists d  gt 0 such that 0 lt x - c lt
  d implies that f(x) - L lt  ?.

Choosing ?  gt 0, means we want the distance of
between f(x) and L to be less than ? and this
distance is a small positive number so this can
restated as given ? gt 0
So we can find delta gt 0 given ? and this can
restated as there exists d  gt 0

• So that the distance between x and c is less than
  d but with x does not equal c (we get really
  close but not there or arbitrarily close)
• The distance between x and c is x c and this
  amount has to be less than d so thats why we
  have x c lt d
• If x does not equal c then x c does not
  equal zero
• So then x c is nonnegative which means x
  c gt 0
• So when we put that all together we get 0 lt
   x - c lt  d

• The distance from f(x) to L is f(x) L and
  this distance has to be less than ?
• So that implies that f(x) - L lt  ?.

5
Example

• Find d that corresponds to ? 0.01 from the
  definition of a limit
• The limit of f(x) as x approaches c is L if and
  only if, given ?  gt 0, there exists d  gt 0 such
  that 0 lt x - c lt  d implies that f(x) - L lt 
  ?.
• Given the and ? 0.01
  we want to find the corresponding d
• So 3x2-12 lt 0.01 and we solve backwards for
  what d has to be
• -0.01 lt 3x2-12 lt 0.01
  property of abs vales
• 11.99 lt 3x2 lt 12.01 add
  12 to both sides
• 3.99666 lt x2 lt 4.00333 divide by 3
  both sides
• 1.99916 lt x lt 2.00083 take the
  square root of both sides
• -0.00083 lt x - 2 lt 0.00083 we need a
  small interval so we - 2 from both sides
• We would want to let d be less than -0.00083
  and 0.00083
• We can always choose d to be smaller
• We can simplify our answer to d 0.0008

6
Example

• Find d that corresponds to ? 0.01 from the
  definition of a limit that
• the limit of f(x) as x approaches c is L if and
  only if, given ?  gt 0, there exists d  gt 0 such
  that 0 lt x - c lt  d implies that f(x) - L lt 
  ?.
• Choose to d 0.0008
• We can verify that this works
• Suppose 0 lt x 2 lt 0.0008 b/c c 2
  and d 0.0008
• -0.0008 lt x 2 lt 0.0008
• 1.9992 lt x lt 2.0008
• 3.99680064 lt x2 lt 4.00320064
• 11.99040192 lt 3x2 lt 12.00960192
• Subtract 12 (our limit)
• -0.00959808 lt 3x2 12 lt 0.00960192
• So then 3x2 12 lt 0.00959808 lt ?

7
Source

• The notes, examples, and information used in this
  brief lesson are taken from
• Visual Calculus website (Thanks!)
• Calculus of a Single Variable
• 7th ed Larson Hostetler Edwards