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Limit of a sequence of numbers 1 to 2

  1. Oct 6, 2005 #1
    Sorry for my ignorance, I am learning calculus on my own now and am 17. I understand it, but of course I can't say that since I can't do any of the problems. I am working on factoring right now, which takes practice. And I havent taken trig. But I am interested in it, nevertheless.

    My question is this. When I was in math i always got A's. And science was my good area too.

    I always asked my algebra teacher math questions, namely, caluclus questions
    She told me that there are infinite number of numbers between 1 and 2
    Is this true? Secondly, would this be a correct limit for it?

    Between 1 and 2, there are an infinite amount of numbers.
    1…1.5…1.9…1.99..1.99..1.999..etc
    Lim f(x)
    1 -> ∞ = 2

    thanks
     
  2. jcsd
  3. Oct 6, 2005 #2

    StatusX

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    That limit is wrong. If the function is 1, a constant, then the limit as x->infinity is 1. It's always 1, no matter what x is. I don't really know what you're trying to say.
     
  4. Oct 6, 2005 #3

    lurflurf

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    There are an infinite number as is easy to see
    1+1/n n=2,3,4,5,...
    are an infinite amount of numbers between 1 and two.
    I do not know what your limit means
    let (1,2) be the set of all numbers between 1 and 2
    that is x is in (1,2) is the same as 1<x<2
    the is somthing called a sup which is the smallest number y so that
    x<y for all x so that 1<x<2
    we see that sup (1,2)=2
    because
    is 1<x<2 x<2
    and is y<2
    2-y>0 so there is y<x<2
     
  5. Oct 6, 2005 #4
    what is sup? Is 1<x<2 the same as x>1<2?
    This is very confusing. coudl you try to explain it in an easier way?
    I cant deduce why exactly this is in calculus yet.
     
  6. Oct 6, 2005 #5
    I read calculus for dummies but it never explained it like this..
     
  7. Oct 6, 2005 #6

    lurflurf

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    sup is the least upper bound
    in this case the smallest number larger than all numbers smaller the 2
    sup (1,2)=2
    1<x<2 means x is greater than 1 and less than 2 that is x is between 1 and 2
    among the numbers between 1 and two there are numbers very close to one and two
    1.1,1.01,1.001,1.0001,1.00000000000000000000000000000001
    1.9,1.99,1.999,1.9999,1.99999999999999999999999999999999
    and so on
    so although if 1<x<2
    the smallest number so that x<y is sup (1,2)=2
    that is any number less than 2 will have some number less than two greater than it
    say 1.9999<2
    1.9999<1.99999<2
    I get the feeling you were asking about the largest number less than 2
    there is not one, but as larger and larger numbers less than 2 are considered numbers ever closer to two are seen so we could say that 2 is a limit of the large values nere two
    possible sequences to consider are of the form
    2-r^n where 0<r<1 ie r=.1
    1.9,1.99,1.999,...
    we see that all the numbers in such sequence are between 1 and 2, but the get ever closer to 2
    [tex]\lim_{n\rightarrow\infty}2-r^n=2[/tex]
    for all r 0<r<1

    I still do not understand your limit
    1 cannot approuch any thing as it is a constant and your f(x) is not clear
     
  8. Oct 6, 2005 #7
    Which one is y?
    f(x)?
    Could you explain how x effects y?

    its ok, i have forgot alot of calculus and do not have the fundamental knowledge to do calculus, plus, calculus is hard for everyoee since it is abstract, oh yeah, and what is n?

    Thakns
     
    Last edited: Oct 6, 2005
  9. Oct 8, 2005 #8

    CRGreathouse

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    To answer your first question, I need to make explicit one of your assumptions. This can be true or false depending on what number system you are using. There are an infinite number of numbers between 1 and 2 in [tex]\mathbb{R}[/tex], the real numbers, but only a finite number (0 or 2, depending on whether you include the endpoints) in [tex]\mathbb{Z}[/tex], the integers.

    To phrase your question "mathematically", you're looking for the cardinality (size) of the interval [tex](1,2)[/tex]. This cardinality is finite (0 or 2) in the integers, infinite ([tex]\aleph_0[/tex], as large as the cardinality of the integers) in the rational numbers [tex]\mathbb{Q}[/tex], and infinite ([tex]\mathfrak{c}[/tex], as large as the cardinality of the real numbers) in the real numbers. Yes, there are as many real numbers between 1 and 2 as there are in total! Isn't that crazy?

    For the second question, there is no limit because there is no order. If you pick a particular sequence between 1 and 2 you can find a limit, but there needs to be an order before this even makes sense. So, to answer your question, there is no limit.
     
  10. Oct 9, 2005 #9

    Hurkyl

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    It doesn't make sense to ask for the limit of the numbers between 1 and 2.

    That collection certainly isn't a sequence, in the ordinary sense. That collection isn't a function either.

    So, it doesn't make sense to ask for its limit as if it were a sequence, or as if it were a function.


    Now, there are interesting things you can ask -- you can ask things like:

    "What is the least upper bound of (1, 2)"?
    "What is the greatest lower bound of (1, 2)"?
    "What is the cardinality of (1, 2)"?
    "What is the order type of (1, 2)"?

    Or, you might write a function like

    f(x) = (3/2) + (1/pi) arctan(x)

    whose image is precisely the interval (1, 2), and then you could ask questions like:

    "What is the limit of f(x) as x goes to -infinity?"
     
  11. Oct 12, 2005 #10
    Thanks Hurky. I am just not at that level ...yet..somehow I think it won't be easy even when I am!

    I haven't learned trig yet, or quadratic equations, and am working on factoring, there are different types of factoring which make it harder.
     
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