1.2 finding limits graphically & numerically. after this lesson, you should be able to: estimate...

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 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.


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


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!


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.


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


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


Type 2 Unbounded Behavior – Case 2

Example 9 Given , find x

xf1

)( )(lim0

xfx


)(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


)(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


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

1.2 finding limits graphically & numerically. after this lesson, you should be able to: estimate...

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