1.2 Finding Limits Graphically

1
1.2 Finding Limits Graphically Numerically
2
After this lesson, you should be able to

• Estimate a limit using a numerical or graphical
  approach
• Learn different ways of determining the
  existence of a limit

3
Calculus centers around 2 fundamental problems

1. The tangent line -- differential calculus

2) The area problem -- integral calculus
4

1. The tangent line- differential calculus

5
Problem Graph y f (x) x2 1 How to
interpret the change in y and the change in x?
For example, the rate of change at some point,
say x 2 is considered as average rate of
change at its neighbor.
(2, f(2)), (2.05, f(2.05)) 2.05 2
0.05 f(2.05) f(2) (2, f(2)), (2.04,
f(2.04)) 2.04 2 0.04 f(2.04) f(2) (2,
f(2)), (2.03, f(2.03)) 2.03 2 0.03 f(2.03)
f(2) (2, f(2)), (2.02, f(2.02)) 2.02 2
0.02 f(2.02) f(2) (2, f(2)), (2.01,
f(2.01)) 2.01 2 0.01 f(2.01) f(2)
Change in x
Change in y
6
In general, the rate of change at a single point
x c is considered as an average rate of change
at its neighbor P(c, f(c)) and Q(c h, f(c h))
under a procedure of a secant line when its
neighbor is approaching to but not equal to that
single point, or, in some other format, it can be
interpreted as (c, f(c)) and (c ?x, f(c
?x)) or, the ratio of change in function value
y f(c ?x) f(c) to the change in variable x,
or, c ?x c ?x
7
Slope of secant line is the average rate of
change
8
when ? 0 the will be approaching to a
certain value. This value is the limit of the
slope of the secant line and is called the rate
of change at a single point AKA instantaneous
rate of change.
9
Note Not any function can have instantaneous
rate of change at a particular specified point

x 0
x 0
x 1
x 2
10
2) The area problem- integral calculus
Question What is the area under a curve bounded
by an interval?
Problem Graph
on
11
Similar to the way we deal with the Rate of
Change, we partition the interval with certain
amount of subinterval with or without equal
length. Then we calculate the areas of these
individual rectangles and sum them all together.
That is the approximate area for the area under
the curve bounded by the given interval. If we
allow the process of partition of the interval
goes to infinite, the ultimate result is the area
under a curve bounded by an interval.
Left Height
Right Height
12
Problem Find the area of the graph
on

• Use 4 subdivisions and draw the LEFT HEIGHTS
• 2) Use 4 subdivisions and draw the RIGHT HEIGHTS

13
Introduction to Limits
Limits are extremely important in the development
of calculus and in all of the major calculus
techniques, including differentiation,
integration, and infinite series. Question
What is limit?

• Problem Given function
  , find
• 1) 2) 3)
• 4) 5) 6)
• 7)

14
Introduction to Limits
Even if the students are forbidden by the evil
Mr. Tu to calculate the f (2), the student could
still figure out what it would probably be by
plugging in an insanely close number like
1.99999999999. It is pretty obvious that
function f is headed straight for the point (2,
7) and thats what is meant by a limit.
Now we have some sense of limit and we could give
limit a conceptual description. A limit is the
intended function value at a given value of x,
whether or not the function actually reaches that
value at the given x. A limit is the value a
function intends to reach.
15
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 TRACE then 2 then
ENTER. Youll see that no y value corresponds
to x 2.
16
Introduction to Limits
Even if the students are forbidden by the evil
Mr. Tu to calculate the f (2), the student could
still figure out what it would probably be by
plugging in an insanely close number like
1.99999999999. It is pretty obvious that
function f is headed straight for the point (2,
1/4) and that is what is meant by a limit.
Now we have some sense of limit and we could give
limit a conceptual description. A limit is the
intended function value at a given value of x,
whether or not the function actually reaches that
value at the given x. A limit is the value a
function intends to reach.
17
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 (or height). ?
18
Definition (informal) Limit
If the function f (x) becomes arbitrarily close
to a single number L (a y-value) as x
approaches c from either side, then the limit of
f (x) as x approaches c is L written as

A limit is looking for the height of a curve at
some x c. L must be a fixed, finite number.
One-Sided Limits?
?Height of the curve approach x c from
the RIGHT ?
?Height of the curve approach x c from the
LEFT
19
Definition (informal) of Limit
If
then
(Again, L must be a fixed, finite number.)
20
Note 1) The definition of a function at one
single value may not exist (defined) but it does
not affect we seek the limit of a function as x
approaches to this single value. 2) The
statement as x approaches to b in limit means
that x can approaches b arbitrarily close but
can NOT equal to b. 3) The statement x can
approaches b arbitrarily close but can NOT equal
to b means that x can approaches to b in any way
it wants, such as left, right, or
alternatively. 4) Some of the questions can be
solved by using the 1.1 knowledge.
21
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.
22
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.
23
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.
24
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)
25
Limits ? Graphically Example 1
Theres a break in the graph at x c
L1
Discontinuity at x c
L2
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.
c
L1
Right Limit
L2
Left Limit
Does not exist since L1 ? L2
26
Limits ? Graphically Example 2
Hole at x c
Discontinuity at x c
L
L
Since these two are the same real number, then
the Limit Exists and the limit is L.
c
Right Limit
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.
27
Limits ?Graphically Example 3
No hole or break at x c
Continuous Function
f(c)
f (c)
Right Limit
c
f (c)
Left Limit
Limit exists
f (c)
In this case, the limit exists and the limit
equals the value of f (c).
28
Limits ? Numerically
On your calculator, graph
Where is f(x) undefined?
at
Although the function is not defined at x 0,
we still can find the intended height that the
function tries to reach.
29
Given , find
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.
30
The LIMIT exists Type 1 Plug in the x value
into function to find the limit when the graph of
the function is continuous Example 3 Given
, find
31
The LIMIT exists Type 2 The function is NOT
defined at the point to which x approaches, the
function is discontinuous at that point and the
graph has a hole at that point. Example 4 Given
, find
Although the function is not defined at x 1,
we still can find the intended height that the
function tries to reach.
32
Example 5 Given ,
find
33
The LIMIT exists Type 3 The function is defined
at the point to which x approaches, however, the
function value is quite different from the value
it SHOULD be. The function is discontinuous
at that point and the graph has an extreme or
outlay value at that point. Example 6 Given
, find
Example 7 Given
, find
34
Conclusion When Does a Limit Exist? The
left-hand limit must exist at x c The
right-hand limit must exist at x c The left-
and right-hand limits at x c must be equal
35
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 5 in text page
  51)

Read Example 5 in text on page 51.
36
The LIMIT does NOT exists Type 1
Limits(Behavior) differs from the Right and Left
Case 1
Example 8 Given , find
37
Limit Differs From the Right and Left- Case 1
1
To graph this piecewise function, this is the
TEST 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)
38
The LIMIT does NOT exists Type 2 Unbounded
Behavior Case 2
Example 9 Given , find
Example 10 Given ,
find
39
Unbounded Behavior- Case 2
DNE
Since f(x) is not approaching a real number L as
x approaches 0, the limit does not exist.
40
The LIMIT does NOT exists Type 3 Oscillating
Behavior Case 3
Example 11 Given , find
as
as
as
as
41
Conclusion When Does a Limit NOT Exist? At
least one of the following holds 1) The
left-hand limit does NOT exist at x c 2) The
right-hand limit does NOT exist at x c 3) The
left- and right-hand limits at x c is NOT
equal 4) A function increases or decreases
infinitely (unbounded) at a given x-value 5) A
function oscillates infinitely and never
approaching a single value (height)
42
Limit
Example
43
Example
44
Does the limit of the function need to equal the
value of a function??
Example
45
Important things to note
1) The limit of a function at x c does not
depend on the value of f (c). 2) The limit only
exists when the limit from the right equals the
limit from the left and the value is a FIXED,
FINITE real number! 3) Limits fail to exist
(ask for pictures) 1. Unbounded behavior not
finite 2. Oscillating behavior not fixed3.
fails def of limit
46
Homework
Section 1.2 page 54 1 7 odd, 9 20, 49
52, 63, 65