my homrwork ms 2

8/13/2019 My Homrwork Ms 2

1/19

LIMITPresented by

Teuku Irfan

Syiah Kuala University 2011


2/19

APPROACHING ...


3/19

For example, what is the value of1

1when x=1?

(Notice in the graph there is actually A HOLE at that point).


4/19

Let us substitute "1" for "x" and see the result:


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x

0.5 1.50000

0.9 1.90000

0.99 1.99000

0.999 1.99900

0.9999 1.99990

0.99999 1.99999

... ...


6/19

Now we can see that as x gets close to 1, then

1

1gets close to 2, so we say :

The limit of1

1as x approaches 1 is 2 and it

is written in symbols as


7/19

More Formal

But you can't just say a limit equals some

value because it looked like it was going to.

We need a more formal definition.So let's start with the general idea


8/19

"f(x) gets close tosome limitas x gets

close to some value

From English to MathematicsLet's say it in English first:

If we call the Limit "L", and the value that x gets close to "a" we can say

"f(x) gets close to L as x gets close to a"


9/19

3 24 6

2

x x xf x

x

Continuity

What happens at x= 2?

What is f(2)?

What happens near x = 2?

f(x) is near -3

What happens as xapproaches2?

f(x) approaches -3


10/19

What happens atx= 1?

What happens nearx= 1?

Asxapproaches1, gincreases without bound, or gapproachesinfinity.

Asxincreases without bound, gapproaches0.

Asxapproachesinfinity g approaches0.

ASYMPTOTES

2

1

1g x

x


11/19

ASYMPTOTES

2

1

1g x

x

x x-1 1/(x-1)^2

0.9 -0.1 100.00

0.91 -0.09 123.46

0.92 -0.08 156.25

0.93 -0.07 204.08

0.94 -0.06 277.78

0.95 -0.05 400.00

0.96 -0.04 625.00

0.97 -0.03 1,111.11

0.98 -0.02 2,500.00

0.99 -0.01 10,000.00

1 0 Undefined

1.01 0.01 10,000.00

1.02 0.02 2,500.00

1.03 0.03 1,111.11

1.04 0.04 625.00

1.05 0.05 400.00

1.06 0.06 277.78

1.07 0.07 204.08

1.08 0.08 156.25

1.09 0.09 123.46

1.10 0.1 100.00


12/19

ASYMPTOTES

2

1

1g x

x

x x-1 1/(x-1)^2

1 0 Undefinned

2 1 1

5 4 0.25

10 9 0.01234567901234570

50 49 0.00041649312786339

100 99 0.00010203040506071

500 499 0.00000401604812832

1,000 999 0.00000100200300401

10,000 9999 0.00000001000200030

100,000 99999 0.00000000010000200

1,000,000 999999 0.00000000000100000

10,000,000 9999999 0.00000000000001000

100,000,000 99999999 0.00000000000000010


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14/19


15/19

As xapproaches 1, (5 2x) approaches ?

f(x)within0.08unitsof3

xwithin0.04unitso

f1

f(x)within0.16units

of3

x

within0.08unitsof

1

0.90 3.20

0.91 3.18

0.92 3.16

0.93 3.14

0.94 3.12

0.95 3.10

0.96 3.080.97 3.06

0.98 3.04

0.99 3.02

1.00 3.00

1.01 2.98

1.02 2.96

1.03 2.94

1.04 2.92

1.05 2.90

1.06 2.88

1.07 2.86

1.08 2.84

1.09 2.82

x 5 2x


16/19

Limit Theorems

Almost all limit are actually found by substituting the

values into the expression, simplifying, and coming up

with a number, the limit.

The theorems on limits of sums, products, powers, etc.

justify the substituting.

Those that dont simplify can often be found with moreadvanced theorems such as L'Hpital's Rule


17/19

THE AREA PROBLEM

21

0

1

3

h x x

j x

x

x


18/19

THE AREA PROBLEM

2

2 2 2 2

22 2

1

22 2

1

length 1

3 1 2width

x-coordinates = 1,1 ,1 2 ,1 3 , ...,1

Area 1 1

32Area = lim 1 1

3

n n n n

n

n n

i

n

n nn

i

h x j x x

b a

n n n

n

i

i


19/19

THE AREA PROBLEM

2

2 3

2 3

2 22 2 2 2 4 2 4

1 1 1 1

8 84 2

1

1 2 1

6

1

8 84

83

32

1

3

2

1

lim 1 1 lim 2 lim lim

lim lim lim

lim lim lim

4 4

1

n n n n

n n n n n n ni n n ni i i i

n n nn n n

n n n

n

i

n

n n n

n

n n

i

n

n

n

i

i

i i

n

i

i

my homrwork ms 2

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