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
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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.
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.
What is a limit?
A limit is the
intended
height
of a function.
How do you determine a function’s height?
Plug an
x
-value
into the function to see how high it will be.
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.
Limit
Notation
Video Clip from Calculus-Help.com
When Does a Limit Exist?
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.
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.
One
Sided Limits
Right-hand Limit
: the 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.
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.
General Limit
A
general limit exists on
f
(
x
) when
x = c
, if the left- and right-hand limits are both equal there.
Finding Limits
Finding Limits
Finding Limits
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.
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