Limits Numerically

1
Limits Numerically

• Warm-Up What do you think the following limit
  equals? If you are unsure at least recall what a
  limit is and see if that helps direct you.

The height of the line y2 is always 2, so the
intended height or where it is heading
towards is always going to be 2!!
2
Objectives

• To determine when a limit exists.
• To find limits using a graphing calculator and
  table of values.
• TS Explicitly assessing information and drawing
  conclusions.

3
What is a limit?

• A limit is the intended height of a function.

4
How do you determine a functions height?

• Plug an x-value into the function to see how high
  it will be.

5
Can a limit exist if there is a hole in the graph
of a function?

• Yes, a limit can exist if the ultimate
  destination is a hole in the graph.

6
Limit Notation
The limit, as x approaches 2, of f (x) is
4. or The limit of f (x), as x approaches 2, is
4.
7
Video Clip fromCalculus-Help.com

• When Does a Limit Exist?

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When does a limit exist?

• A limit exists if you travel along a function
  from the left side and from the right side toward
  some specific value of x, and
• As long as that function meets in the middle, as
  long as the heights from the left AND the right
  are the same, then the limit exists.

9
When does a limit not exist?

• A limit will not exist if there is a break in the
  graph of a function.
• If the height arrived at from the left does not
  match the height arrived at from the right, then
  the limit does not exist.
• Key Point If a graph does not break at a given
  x-value, a limit exists there.

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One Sided Limits
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Right-hand Limitthe height arrived at from the
right

• Read as The limit of f (x) as x approaches 4
  from the right equals 2.
• This means x approaches 4 with values greater
  than 4.

12
Left-hand Limit the height arrived at from the
left

• Read as The limit of f (x) as x approaches 4
  from the left equals 1.
• This means x approaches 4 with values less than 4.

13
General Limit

• A general limit exists on f (x) when x c, if
  the left- and right-hand limits are both equal
  there.

• Mathematic Notation
• In other words
• f (x) ? L as x ? c

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Finding Limits
7
7
7
If a function approaches the same value from
both directions, then that value is the limit
of the function at that point.
x
g (x)
x
g (x)
.9
1.1
6.71
7.31
.99
1.01
6.9701
7.0301
.999
1.001
6.997
7.003
15
Finding Limits
DNE or NL
3
3
If the Left-hand limit and the Right-hand limit
are not equal, the general limit does not
exist.
x
h (x)
x
h (x)
1.1
.9
3.1
2.9
1.01
.99
3.01
2.99
3.001
1.001
.999
2.999
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Finding Limits
DNE or NL
NL
NL
x
j (x)
x
j (x)
If either the Left-hand limit, Right-hand
limit, or both do not exist, the general limit
will not exist.
2.9
3.1
44.1
56.1
2.99
3.01
494
506.01
2.999
3.001
4994
5006
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Conclusion

• A limit is the intended height of a function.
• A limit will exist only when the left- and
  right-hand limits are equal.
• A limit can exist if there is a hole in the
  graph.
• A limit will not exist if there is a break in the
  graph.