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Section 1.5: Finding Limits Graphically and Numerically
1. Graphical Limits:
Notation: Recall from previous experiences,
and
means
means
Given the graph of a piecewise function ( ), find the value of ( ) as
a)
b)
c)
2. Notation
The notation
i.e.,
or
A two-sided limit exists if
Thus,
From the above,
( )
a)
b)
( )
c)
( )
is used when writing one-sided limits,
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Ex.) Use the previous graph to find the value of the limit
( )
1.
( )
2.
3.
( )
Important Note!!
3. Numerical Limits:
If a graph is not available, or the function is difficult to graph, construct a table of
values. Select values “close” to , (i.e.,
) and evaluate.
Ex.) Find
Thus,
√
√
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4. Limits that Fail to Exist
There are three different scenarios where we say the limit does not exist.
a)
b)
c)
5. Formal Definition of a Limit
Let ( ) be defined on an open interval containing (except possibly at
be a real number. Then
( )
if
|
means for each
|
then | ( )
there exists a
|
such that
itself) and let
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Ex.) Use the
definition to prove each limit exists
1)
(
)
2)
(
)