1even i m not convinced equal limits never tell equal value
Which of these is greatest
.9 (0.9999999999999999999................)
1.01 (1.00000000000000.................0001)
1
??
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70 Answers
1Oh well, I suppose I must agree with you, because many renowned websites are of the same opinion, including Wikipedia. But deep inside, I'm still not a tad bit convinced. Anyway, let's end this debate right here, as its not going to end otherwise. :)
1uh-oh..limits again!!
well, vicky, wat if n tends to infinity?? you hav 2 agree this time that as n tends to infinity, n-1 tends to n.....
1shreyan, your post #47 has a flaw.
let there be n 9s in 0.999999...............
then 9.99999................ will have (n-1) 9s after the decimal.
So, when you subtract, you'll have
9x = 9.99999999............9 - 0.999999999.............9
(n-1) times n times
=>9x = 8.999999................91
(n-1) times
=>x = 0.9999999.............9
n times
which obviously is what you started out with in the first place. Correct?
1if anyone wants proof for
0.9(bar) = 1
i have dozens of them........
1well vicky, if u r not convinced wid the limit method of class 11 n 12, lets do it using class 9-10 method....
let 0.9(bar) = x .............................................(1)
then 10x = 9.9(bar).......................................(2)
sub (1) frm (2)
9x = 9
=> x=1
similarly, 1.0(bar)1 = 1
62did you do these problems find
x in fraction form where x=.1666666666666666...............
we used the same trick
10x=1.66666666666666666666666666
substracting
9x=1.5
x=1.5/9 = 3/18=1/6
1Looks like this is one question where we must agree to disagree ;). sorry but I'm still not convinced sir. I'm still open to explanations though.
62vicky, are u in class XI or XII.?
you are a relatively new user here. so sorry that I dont know!
62is lim f(x) = f(a)??
x→a
this is true for continuous functions :)
1I'm still not getting the point. As for your question, I'd put it this way:
is lim f(x) = f(a)??
x→a
And, sir, I wouldn't mind a pure mathematics based explanation as long as I get this point cleared.
62Vicky,
I dont want to go into pure mathematics intrying to explain this to you
But what is the equality of 2 numbers?
when do we say that x>y or x<y or x=y?
1naturally <1, since (1-h) still tends to 1- if h tends to 0+, it has nothing to do with the value of (1-h) at h exactly equal to 0. limit of f(x) as x→a has nothing to do with f(a), right? if I ask
lim x2-4
x→2 x-2
it has nothing to do with the value of f(x) at x exactly equal to 2 or f(2), since f(2) is not defined in this case.(f(x) being that function).
1Ok just adding one random doubt that came to my mind...
what is
\left\{1.\bar{9}\right\}
0 or 1 ?
49though the old one but i have a confusion....
when we say, lim (y)=b
x->a
means as x->a, y->b.....not y=b isnt it?? so???[4]
agree with other methods but still this one hurts me...
1(1.00....1) >1 > (0.999...9)(1.00...1)
let. a=1, b=0.000...1
then, (0.999...9)(1.00...1)= (a+b)(a-b)=a.a - b.b =1 - 0.000...1, which is less than 1.
1WHOA!!! wiki's gud!!
and after all this discussion, here's quoting wiki:
1ur welcome, vicky!!!
and ya..ur proof is also very elegant!!! lemme hav a luk at wiki too...
1Hold on... lets halt our mathematical horses right there ;) Here's one simple yet infallible proof (Source: Wikipedia):-
We know that
1/3 = 0.3
Multiply both sides by 3,
1 = 0.9
Celestine, you can see very clearly this proof has no flaw whatsoever.
And if you're interested in a host of other proofs, check out this link:
http://en.wikipedia.org/wiki/0.99999
The article in the above link has got one interesting thing to say:
"There are no non-zero infinitesimals." Sounds a bit weird, but its apparently been proved and is called "the Archimedean property of the standard real numbers".
Anyway, its for you to decide now, but the proof involving 1/3 has completely convinced me. And, Nishant sir, I must thank you for so patiently explaining things to everyone :). Thanks a lot. And thanks to shreyan and philip too :). This community is great, I must say.
62so what if i said that
\large \lim_{n\rightarrow \infty}{0.9999.... n \text{times}}
will it be <1 or =1?
11Why this debate???? There might be one millionth of a hair's width between 0.9 and 1.....But still 1 > 0.9
Similarly 1.01 > 1 [4]
9i jus said that
[.99999999999999999] = lt h→0- [1+h] and thats true isnt it ???