Question

i saw a proof in wikipedia about 0.9999999........... the proof says that the above is equal to 1 . now if f(x) = [x] , where [ ] represents greatest integer function would the limit exist at x=1

Answer 1

in fact .9 repeating IS 1

Answer 2

do u want a proof?

Answer 3

oh that is not what you are asking i see

Answer 4

and the answer is no the limit at 1 of the greatest integer function does not exist

Answer 5

yes..certainly..the limit at an integer is nonexistent

Answer 6

the greatest integer function has a jump at 1.

Answer 7

then if x = 0.99999999999........
would d limit exist then

Answer 8

no it doesnt exist

Answer 9

because 0.99999999999999999999999 IS infact 1

Answer 10

i see the problem. you are assuming that .999... is somehow less than one. it is if you truncate, but if you write .9 repeating that number is 1 and no other

Answer 11

yes satellites right..0.9 bar is infact one

Answer 12

and no other

Answer 13

if technically seeing it approaches to 1

Answer 14

which in fact is a very good explanation of why the limit does not exist!
[.9] = [.99] = [.999]=[.9999]=....=0

Answer 15

no matter how far out you go, as long as you stop then you will get 0 for the greatest integer function. but evaluated at .9 repeating = 1 you get 1

Answer 16

the real question is what does it mean to say .9 repeating =1, and it really comes down to "what does .9 repeating actually mean?"

Answer 17

yes sir ....

Answer 18

here is an example you are probably more familiar with. most people probably have seen that
\[\frac{1}{3}=.333...\] yes?

Answer 19

see if u convert 1/9 into a decimal...u get 0.111111111111111111
so
1/9= 0.111111111111......
multiply both sides by 9
1/9 x 9 = 0.1111111111111111111111111111111111....... x 9
1= 0.999999....

Answer 20

@him this is an explanation, but it begs the question of what .1 repeating actually means. after all if i cannot parse .1 repeating i cannot multiply it by 9, since i do not have enough time

Answer 21

.1 repeating actually means 1/9..u can see that in the method to convert a repeating decimal into a rational number representation...

Answer 22

point being that i do not have enough time to divide 1 by 9! i will be doing it forever. i see the pattern but that does not explain the meaning of .1 repeating or .3 repeating or .9 repeating

Answer 23

x = 0.111...
10x = 1.111...
subtract one from two...
9x = 1
x = 1/9
hence
0.111...=1/9

Answer 24

i know the proof sir . i actually want to ask will the limit exist at x = 0.9999999999.....
when f(x) = [x]

Answer 25

no....

Answer 26

sir mat bula yaar..itna buddha bhi nahin hoon

Answer 27

i can make no sense out of writing an infinite number of decimals without resorting to a limit statement.
\[\bar{.1}=lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{1}{10^k}\]

Answer 28

\[\bar{.9}=lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{9}{10^k}\]

Answer 29

its a GP

Answer 30

i cannot add subtract multiply or divide an infinite number of decimals. i have to say precisely what i mean by them. this requires work, although the "trick" of multiplying and subtracting is what we will do. but the understanding means i have to kn ow what a limit is and how i can say .9 repeat in IS 1

Answer 31

both my proof and urs are on wikipedia....it says its been proven by methods of varying mathematical rigour....

Answer 32

sure of course. you have to begin where you are

Answer 33

if i want to convince someone that .9 repeating is 1, that is exactly what i would say, what you said!

Answer 34

but that begs the question of what .9 repeating MEANS. that is my point.

Answer 35

it means one...
"In mathematics, the repeating decimal 0.999..., which may also be written as 0.9, or 0.(9), denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. "

Answer 36

this says nothing other than .9 repeating is 1. it does not explain what it means write in infinite repeating decimal

Answer 37

its just a fraction which when evaluated gives a nonterminating repeating sequence of digits..dont get technical now,.. : )

Answer 38

there is nothing special about decimal fractions. you can use base anything you like. so for example i can use base 2 and then .1 repeating = 1
or i can use base 3 and then .2 repeating = 1

Answer 39

or use base 4 and get .4 repeating =1 you get the picture

Answer 40

that was wrong. in base 4 it is .3 repeating = 1

Answer 41

so in base 4
.1 repeating would mean
\[\bar{.1}=lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{1}{4^k}\]

Answer 42

true....

Answer 43

so now wht do u want to prove?

Answer 44

only point i am trying to make, a simple point really, is that we need to understand what we mean by a repeating decimal. it is a limit of a sum. we are so used to writing numbers in base ten that we forget that they have a specific meaning.

Answer 45

right....

Answer 46

even terninating decimals right? i have to remind myself that
\[.251=\frac{2}{10}+\frac{5}{100}+\frac{1}{1000}\]

Answer 47

so for greatest integer function we see that
\[ f(\bar{.9})=f(lim_{n\rightarrow \infty}\sum_{k=1}^n\frac{9}{10^k})=1\]

Answer 48

is f(x) 1 or is the limit 1?

Answer 49

but
\[f(\sum_{k=1}^n\frac{9}{10^k})=0\] for all finite n

Answer 50

i think the limit is 1...so for all mathematical puposes 0.999.. is 1

Answer 51

yes i wrote limit. take the limit, get 1. evaluate greatest integer function, get 1.
but until you pass to the limit you do not get 1 and therefore the greatest integer function will be 0

Answer 52

so just before it the limit is 0, bt just after it it is 1..so it doesnt exist

Answer 53

and thus not continuous of course. picture tells us that

Answer 54

yes

Answer 55

well there is no "just before the limit". there is the limit and there is everything else. but yes, until you take the limit you get something less than one. and once you take the limit you get 1

Answer 56

by just before i mean the left hand limit..r u phd in maths?

Answer 57

btw if the greatest integer function were continuous you would be able to "take the limit outside the function" as they say. which of course you cannot do

Answer 58

actually no i am just a well trained dog. that is the beauty of the internet
http://www.unc.edu/depts/jomc/academics/dri/idog.jpg

Answer 59

nice...lol