Finding Limits Graphically

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Finding Limits Graphically Numerically

• Section 1.2

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After this lesson, you should be able to

• estimate a limit using a numerical or graphical
  approach
• determine the existence of a limit

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Introduction to Limits
The function
is a rational function.
Graph the function on your calculator.
If I asked you the value of the function when x
4, you would say
What about x 2?
Well, if you look at the function and determine
its domain, youll see that . Look
at the table and youll notice ERROR in the y
column for 2 and 2.
On your calculator, hit ? then ? then ?. Youll
see that no y value corresponds to x 2.
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Continued
Since we know that x cant be 2, or 2, lets see
whats happening near 2 and -2
Lets start with x 2.
Well need to know what is happening to the right
and to the left of 2. The notation we use is
read as the limit of the function as x
approaches 2.
?In order for this limit to exist, the limit from
the right of 2 and the limit from the left of 2
has to equal the same real number. ?
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Right and Left Limits
To take the right limit, well use the notation,
The symbol to the right of the number refers to
taking the limit from values larger than 2.
To take the left limit, well use the notation,
The - symbol to the right of the number refers to
taking the limit from values smaller than 2.
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Right Limit?Numerically
The right limit
Look at the table of this function. You will
probably want to go to TBLSET and change the ?
TBL to be .1 and start the table at 1.7 or so.
As x approaches 2 from the right (larger values
than 2), what value is y approaching?
You may want to change your ? TBL to be something
smaller to help be more convincing. The table
can be deceiving and well learn other ways of
interpreting limits to be more accurate.
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Left Limit?Numerically
The left limit
Again, look at the table.
As x approaches 2 from the left (smaller values
than 2), what value is y approaching?
Both the left and the right limits are the same
real number, therefore the limit exists. We can
then conclude,
To find the limit graphically, trace the graph
and see what happens to the function as x
approaches 2 from both the right and the left.
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Text
In your text, read An Introduction to Limits on
page 48. Also, follow Examples 1 and 2.
Limits can be estimated three ways Numerically
looking at a table of values Graphically. using
a graph Analytically using algebra OR calculus
(covered next section)
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Limits ? Graphically Example 1
Theres a break in the graph at x c
L1
L2
c
Although it is unclear what is happening at x c
since x cannot equal c, we can at least get
closer and closer to c and get a better idea of
what is happening near c. In order to do this we
need to approach c from the right and from the
left.
Discontinuity at x c
L1
Right Limit
L2
Left Limit
Does not exist since L1 ? L2
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Limits ? Graphically Example 2
Hole at x c
L
c
Right Limit
L
Since these two are the same real number, then
the Limit Exists and the limit is L.
Discontinuity at x c
L
Left Limit
L
L ? f(c)
Note The limit exists but
This is okay!
The existence or nonexistence of f(x) at x c
has no bearing on the existence of the limit of
f(x) as x approaches c.
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Limits ?Graphically Example 3
No hole or break at x c
f(c)
f(c)
Right Limit
c
f(c)
Left Limit
Continuous Function
Limit exists
f(c)
In this case, the limit exists and the limit
equals the value of f(c).
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Limit Differs From the Right and Left- Case 1
1
To graph this piecewise function, this is the ?
menu
0
Limit Does Not Exist
The limits from the right and the left do not
equal the same number, therefore the limit DNE.
(Note I usually abbreviate Does Not Exist with
DNE)
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Unbounded Behavior- Case 2
DNE
Since f(x) is not approaching a real number L as
x approaches 0, the limit does not exist.
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A limit does not exist when

• f(x) approaches a different number from the right
  side of c than it approaches from the left side.
  (case 1 example)
• f(x) increases or decreases without bound as x
  approaches c.
  (The function goes to /- infinity
  as x ? c case 2 example)
• f(x) oscillates between two fixed values as x
  approaches c. (example 5 in text page 51)

Read Example 5 in text on page 51.
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Limits ? Numerically
On your calculator, graph
Where is f(x) undefined?
at
Use the table on your calculator to estimate the
limit as x approaches 0.
Take the limit from the right and from the left
The limit exists and the limit is 2.
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Homework
Section 1.2 page 54 3, 7-15 odd, 19, 49, 53,
63, 65