Post on 01-Jan-2016
1.2 Finding Limits Graphically & Numerically
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
Calculus centers around 2 fundamental problems:
1) The tangent line -- differential calculus
2) The area problem -- integral calculus
1) The tangent line- differential calculus
sec
( ) ( ).
f x x f xAvg rate of change m slope
x
PQ
tan0
( ) ( )limx
f x x f xm
x
Instantaneous rate of change
(Slope at a point)
(2, f(2)), (2.05, f(2.05)) 2.05 – 2 = 0.05f(2.05) – f(2)(2, f(2)), (2.04, f(2.04)) 2.04 – 2 = 0.04f(2.04) – f(2)(2, f(2)), (2.03, f(2.03)) 2.03 – 2 = 0.03f(2.03) – f(2)(2, f(2)), (2.02, f(2.02)) 2.02 – 2 = 0.02f(2.02) – f(2)(2, f(2)), (2.01, f(2.01)) 2.01 – 2 = 0.01f(2.01) – f(2)
ProblemGraph 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.
Change in x Change in y
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 + Δx, f(c + Δx))
or, the ratio of change in function value y
f(c + Δx) – f(c)
to the change in variable x, or, c + Δx – c = Δx
x
cfxcfmchangeofrateAvg
)()(
. sec
Slope of secant line is the “average rate of change”
PQ
x
xfxxfm
x
)()(lim
0tanInstantaneous rate of change
(Slope at a point)
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.
x
x
xfxaf
axa
xfxafm
)()()()(
sec
Note
Not any function can have instantaneous rate of change at a particular specified point1)
2)
3)
4)
xf(x)
1
xxf )(
x = 0
x = 0
2)1()( xxf
2)( xxf
x = 1
x = 2
2) The area problem- integral calculus
Uses rectangles to approximate the area under a curve.
Question: What is the area under a curve bounded by an interval?
Problem
Graph3234.0)( 23 xxxxfy
]15,1[],2,6[ yx
on
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.
Uses rectangles to approximate the area under a curve.
Left Height
Right Height
1) Use 4 subdivisions and draw the LEFT HEIGHTS
2) Use 4 subdivisions and draw the RIGHT HEIGHTS
Problem Find the area of the graph
3234.0)( 23 xxxxfy
]15 ,1[ ],2 ,6[ yxon
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)
)2(f )99.1(f )999.1(f
)9999.1(f )01.2(f )001.2(f
)0001.2(f
32)( xxf
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 that’s 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.
Introduction to Limits
Introduction to LimitsThe function
2
2( )
4
xf x
x
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
1( )
8f x
What about x = 2?
Well, if you look at the function and determine its domain, you’ll see that . Look at the table and you’ll notice ERROR in the y column for –2 and 2.
2,2x
On your calculator, hit “TRACE” then “2” then “ENTER”. You’ll see that no y value corresponds to x = 2.
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.
Introduction to Limits
Continued…Since we know that x can’t be 2, or –2, let’s see what’s happening near 2 and -2…
Let’s start with x = 2.
We’ll need to know what is happening to the right and to the left of 2. The notation we use is:
22
2lim
4x
x
x
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).
One-Sided Limits: Height of the curve approach x = c
from the RIGHT
Height of the curve approach x = c from the LEFT
Definition (informal) Limit
lim ( )x c
f x L
1)(lim Lxfcx
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.
2)(lim Lxfcx
Lxfxfcxcx
)(lim)(lim
Lxfcx
)(lim
Definition (informal) of Limit:
If then
(Again, L must be a fixed, finite number.)
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.
Right and Left LimitsTo take the right limit, we’ll use the notation,
22
2lim
4x
x
x
The + symbol to the right of the number refers to taking the limit from values larger than 2.
To take the left limit, we’ll use the notation,
22
2lim
4x
x
x
The – symbol to the right of the number refers to taking the limit from values smaller than 2.
Right LimitNumerically
The right limit:
22
2lim
4x
x
x
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?
1
4
You may want to change your TBL to be something smaller to help be more convincing. The table can be deceiving and we’ll learn other ways of interpreting limits to be more accurate.
1
4
Left LimitNumerically
The left limit:
22
2lim
4x
x
x
Again, look at the table.
As x approaches 2 from the left (smaller values than 2), what value is y approaching?
1
4
1
4
Both the left and the right limits are the same real number, therefore the limit exists. We can then conclude,
22
2 1lim
4 4x
x
x
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.
TextIn 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)
lim ( )x c
f x
Limits Graphically: Example 1
c
L2
L1 Discontinuity at x = c
There’s a break in the graph at x = 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.
lim ( )x c
f x
lim ( )x c
f x
lim ( )x c
f x
L1
L2
Does not exist since L1 L2
Right Limit
Left Limit
Limits Graphically: Example 2
c
Hole at x = cDiscontinuity at x = c
lim ( )x c
f x
lim ( )x c
f x
lim ( )x c
f x
L
L
Right Limit
Left Limit
L
Since these two are the same real number, then the Limit
Exists and the limit is L.
Note: The limit exists but lim ( )x c
f x
L f (c)
L
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.
This is okay!
Limits Graphically: Example 3
c
Continuous Functionf(c)
No hole or break at x = c
lim ( )x c
f x
lim ( )x c
f x
lim ( )x c
f x
f (c)
f (c)
Right Limit
Left Limit
f (c)
In this case, the limit exists and the limit equals the value of f (c).
Limit exists
Limits Numerically
On your calculator, graph( )1 1
xf x
x
Where is f(x) undefined?
at 0x
Although the function is not defined at x = 0, we still can find the “intended” height that the function tries to reach.
11)(
x
xxf
Use the table on your calculator to estimate the limit as x approaches 0.Take the limit from the right and
from the left:1
lim ( )x
f x
2
1lim ( )x
f x
2The limit exists and the limit is 2.
Given , find 11
)(
x
xxf
11lim)(lim
00
x
xxf
xx
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 43)( 5 xexf )(lim
2xf
x
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 1
1)(
3
x
xxf
1
1lim)(lim
3
11
x
xxf
xx
1
1)(
3
x
xxf
Although the function is not defined at x = 1, we still can find the “intended” height that the function tries to reach.
Example 5 Given , find 4
35)(
x
xxf
4
35lim)(lim
44
x
xxf
xx
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 )(lim
0xf
x
Example 7 Given , find
0 ,2
0 ,1)(
x
x xf
1 2,
1 ,1)(
3 xx
x xf )(lim
1xf
x
Conclusion – When Does a Limit Exist?The left-hand limit must exist at x = cThe right-hand limit must exist at x = cThe left- and right-hand limits at x = c must be equal
2
9)(lim)(lim
33
xfxf
xx
definednot is )3(f2
9)(lim
3
xf
x
A limit does not exist when:
1. f(x) approaches a different number from the right side of c than it approaches from the left side. (case 1 example)
2. f(x) increases or decreases without bound as x approaches c. (The function goes to +/- infinity as x c : case 2 example)
3. 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.
The LIMIT does NOT exists
Type 1 Limits(Behavior) differs from the Right and Left – Case 1
Example 8 Given , find 1
1)(
2
x
xxf )(lim
1xf
x
1 if ,
1
1
1 if ,1
1
1
1)(
2
xx
xx
x
xxf
x
y
2
1
1
1lim)(lim
11
xxf
xx
2
1
1
1lim)(lim
11
xxf
xx
Limit Differs From the Right and Left- Case 1
1lim ( )x
f x
1lim ( )x
f x
1lim ( )x
f x
Limit Does Not Exist
11
limx
x
2
1lim 1x
x
0
2
, if 1( )
1, if 1
x xf x
x x
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)
To graph this piecewise function, this is the TEST menu
The LIMIT does NOT exists
Type 2 Unbounded Behavior – Case 2
Example 9 Given , find x
xf1
)( )(lim0
xfx
Example 10 Given , find
)(lim1
xfx4)1(
1)(
xxf
Unbounded Behavior- Case 2
2
1)(
xxf
20
1limx x
20
1limx x
2
1lim
x0x
DNE
Since f(x) is not approaching a real number L as x approaches 0, the limit does not exist.
x
y
Example 11 Given , find
)(lim0
xfxx
xf1
sin)(
1
10
22
rxk
as k
1
10
22
lxk
as k
1
10 0lim ( ) lim ( ) 1
rr
x xf x f x
1
10 0lim ( ) lim ( ) 1
ll
x xf x f x
The LIMIT does NOT exists
Type 3 Oscillating Behavior – Case 3
2
10
22
rxk
as k
2
20 0lim ( ) lim ( ) 1
rr
x xf x f x
2
10
22
lxk
as k
2
20 0lim ( ) lim ( ) 1
ll
x xf x f x
Conclusion – When Does a Limit NOT Exist?At least one of the following holds
1) The left-hand limit does NOT exist at x = c2) The right-hand limit does NOT exist at x = c3) The left- and right-hand limits at x = c is NOT
equal4) A function increases or decreases infinitely
(unbounded) at a given x-value5) A function oscillates infinitely and never
approaching a single value (height)
Limit:
f (2) =
)(lim2
xfx
)(lim4
xfx
)(lim2
xfx
)(lim2
xfx
)(lim xfx
)(lim4
xfx
)(lim4
xfx
f (4)=
)(lim xfx
Example:
)(lim0
xfx
)(lim4
xfx
)(lim0
xfx
)(lim0
xfx
)(lim xfx
)(lim4
xfx
)(lim4
xfx
f (4) =
)(lim xfx
f (0) =
)(lim3
xfx
)(lim6
xfx
)(lim3
xfx
)(lim3
xfx
)(lim xfx
)(lim6
xfx
)(lim6
xfx
f (6)=
)(lim xfx
f (3) =
Example:
)(lim2
xfx
)(lim5
xfx
)(lim2
xfx
)(lim2
xfx
)(lim xfx
)(lim5
xfx
)(lim5
xfx
f (5)=
)(lim xfx
f (2) =
Does the limit of the function need to equal
the value of a function??
Example:
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)(lim)(lim xfxf
cxcx
Homework
Section 1.2: page 54 #1 ~ 7 odd, 9 ~ 20, 49 ~ 52, 63, 65