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.
If I asked you the value of the function when x
4, you would say ______
What about when 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
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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. We
can also set Indpnt to Ask and enter our own
values close to 2.
As x approaches 2 from the right (larger values
than 2), what value is y approaching?
x 2.2 2.1 2.01 2.001 2
f(x)
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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?
x 1.8 1.9 1.99 1.999 2
f(x)
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The Limit?Numerically
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.
You can ZOOM IN to see x values very close to 2.
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Another Example Finding Limit Numerically
On your calculator, graph
Where is f(x) undefined?
Use the table on your calculator to estimate the
limit as x approaches 0.
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One more Example
EXAMPLE Find the numerically
and verify graphically.   Answer
Use Radian mode.
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Methods for finding limits
Notation roughly
translates into The value of f(x), as x
approaches c from either side, becomes close to
L.
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
L1
Theres a break in the graph at x c
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
Right Limit
Left Limit
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Limits ? Graphically Example 2
Hole at x c
L
c
Right Limit
Discontinuity at x c
Left Limit
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)
Right Limit
c
Left Limit
Continuous Function
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
To graph this piecewise function, this is the ?
menu
The limits from the right and the left do not
equal the same number, therefore the limit
____________________________________.
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Unbounded Behavior- Case 2
Since f(x) is not approaching a real number L as
x approaches 0, the limit does not exist.
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Oscillating Behavior- Case 3
Look at the graph of this function (in radian
mode).
Since f(x) is oscillates between 1 and 1 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. (case 3 example)

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Homework
Section 1.2 page 54 3, 7-15 odd, 19, 49, 53,
63, 65