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Proof of limits

  1. Jun 30, 2015 #1
    How do you prove

    1/x = 0 if x= infinite
    and
    1/x = infinite if x=0

    No need to use formal language, please just explain what the variables are for the quantifiers, or what you mean by for all and there exists if you are gracious enough.
     
  2. jcsd
  3. Jun 30, 2015 #2

    pwsnafu

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    First, if we are talking about the real numbers then they are not true, hence you cannot prove them.
    Second, don't use "infinite" as a noun. It's "infinity".
    Third, ##\frac{1}{ \infty} = 0## on the extended real numbers by definition. There is nothing to prove.
    Fourth, ##\frac{1}{0}## is undefined on both the real numbers and extended real numbers.

    Edit: Wait are you asking for a proof that ##\lim_{x\to\infty} \frac{1}{x} = 0##?
     
  4. Jun 30, 2015 #3
    My professor told me the exact opposite about infinity.

    What do you mean extended real numbers and what do you mean by definition
     
  5. Jun 30, 2015 #4
    Actually I think I used infinity as an adjective, and my professor stopped me and was like infinite and he yelled at me.

    How can infinity be a noun if it means increases without bound
     
  6. Jun 30, 2015 #5

    PeroK

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    "Infinity" is a noun, but (like many nouns) can be use as an "attributive" noun, where it acts like an adjective. E.g. an infinity simulator.

    "Infinite" is an adjective, but (like many adjectives) can be used as an "adjectival" noun. E.g. we are discussing the infinite here.
     
  7. Jun 30, 2015 #6
    These limits do not exist because you can always pick a higher number , y, such that 1/x > 1/y >0.
    and
    and you can pick a smaller number y such that infinity > 1/y > 1/x
     
  8. Jun 30, 2015 #7

    pwsnafu

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    Nouns can represent concepts. "Infinity" means the concept of not being finite.

    If we are talking about real-valued functions, then ##\lim_{x\to0} \frac{1}{x}## does not exist. However, ##\lim_{x\to\infty}\frac{1}{x}## does exist and is equal to zero.

    I've never seen infinite used as a noun in mathematics discourse. Philosophy yes, but never in mathematics.
     
  9. Jun 30, 2015 #8

    HallsofIvy

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    Since what you say your professor did was outrageous and he is not here to defend himself, I am going to ignore that. I suspect you are misunderstanding.

    To talk about "1/0" and "1/infinity" as if they were numbers is simply incorrect. But I notice that this thread is titled "Proof of limits". That's a completely different matter! The "limit as x goes to 0" or the "limit as x goes to infinity" is not saying "x is 0" or "x is infinity". "Going to infinity" is simply "shorthand" for "getting larger and larger without bound.

    Since you talk about limits, I assume that you have seen the definition: We say that "the limit of f(x), as x goes to a, is L" or [itex]\lim_{x\to a} f(x)= L[/itex], if and only if "given any [itex]\epsilon> 0[/itex] there exist [itex]\delta> 0[/itex] such that if [itex]|x- a|< \delta[/itex] then [itex]|f(x)- L|< \epsilon[/itex]. Roughly speaking that means that we can get f(x) as close to L as we please by making x close enough to a.

    For the symbol "infinity", which is not really a number, we interpret it as "getting larger and larger" [itex]\lim_{x\to \infty} f(x)= L[/itex] is defined as "given any [itex]\epsilon> 0[/itex] there exist X such that if x> X then [itex]|f(x)- L|< \epsilon[/itex]. For [itex]\lim_{x\to a} f(x)= \infty[/itex] we have "given any M, there exist [itex]\delta> 0[/itex] such that if [itex]|x- a|< \delta[/itex] then f(x)> M.

    To prove that [itex]\lim_{x\to \infty} \frac{1}{x}= 0[/itex], I would note that [itex]|f(x)- L|= |\frac{1}{x}- 0|= \frac{1}{x}[/itex] since x going to infinity means x must be positive. So I want to get [itex]\frac{1}{x}< \epsilon[/itex] which is the same as [itex]\frac{1}{\epsilon}< x[/itex]. For any [itex]\epsilon> 0[/itex], [itex]\frac{1}{\epsilon}[/itex] exists. Take M in the definition to be [itex]\frac{1}{\epsilon}[/itex].

    To prove [itex]\lim_{x\to 0} \frac{1}{x}= \infty[/itex], I would note that [itex]f(x)= \frac{1}{x}> M[/itex] is the same as [itex]x> \frac{1}{M}[/itex]. So, given such an M, I would take [itex]\delta= \frac{1}{M}[/itex]. Then if [itex]|x- 0|< \delta = \frac{1}{M}[/itex], [itex]\frac{1}{x}> M[/itex].

    Notice that there is no use of a "number" infinity in either of those.
     
    Last edited by a moderator: Jun 30, 2015
  10. Jun 30, 2015 #9
    What does for all epsilon>0 there exists delta > 0 mean?

    Is the for all epsilon just a qualifier for delta, like you are defining what delta can be?


    For the symbol "infinity", which is not really a number, we interpret it as "getting larger and larger" lim x→∞ f(x)=L \lim_{x\to \infty} f(x)= L is defined as "given any ϵ>0 \epsilon> 0 there exist X such that if x> X then |f(x)−L|<ϵ


    How do you go from |x-a| < delta to x>X

    Thank you very much I always wanted to understand this stuff.
     
  11. Jun 30, 2015 #10
    Idk man i confused not being real with not being a noun.


    I just don't know what a limit is, you treat it like x=X but x does not equal X, that's probably what I mean when I said the limits do not exist.


    Yea bro that was my philosophy teacher who said that I take philosophy classes because they are easier than math classes, everyone who gets a question wrong in math should switch majors to philosophy because its easier.
     
  12. Jun 30, 2015 #11

    jbriggs444

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    Let's start with something easier. "for all epsilon > 0, epsilon2 > 0" Can you figure out what that means? Is it a meaningful statement? Is it true or is it false?
     
  13. Jun 30, 2015 #12
    if all numbers in this problem are positive, the square of all numbers in the problem will be positive. that's how I interpret that statement

    for all epsilon > 0, there exists delta > 0. ---- How is that meaningful, it says all the numbers in the problem are positive and there is a positive number that we call delta, who cares about for all. Please explain
     
  14. Jun 30, 2015 #13

    jbriggs444

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    That is not correct. The statement "for all epsilon > 0, epsilon2 > 0" makes no mention of all the numbers in "the problem". Nor does it take the form of an if then statement.

    It is an assertion that every number that is greater than zero has a square that is greater than zero.
     
  15. Jun 30, 2015 #14

    Mark44

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    "If - then" is implied. One could read the above as saying, "if ##\epsilon## is any positive number, then ##\epsilon^2## is positive."
     
  16. Jun 30, 2015 #15

    jbriggs444

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    That is not correct. The for all is outside the if. You are simply mistaken.
     
  17. Jun 30, 2015 #16

    Mark44

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    You're going to have to do better than that.

    How does "for all epsilon > 0, epsilon2 > 0" differ in meaning from "if ##\epsilon## is any positive number, then ##\epsilon^2## is positive." ?
     
  18. Jun 30, 2015 #17
    Why can't you just say; x^2 > 0, where x > 0.

    So for all epsilon > 0, there exists delta > 0 means for every number greater than 0 there exists a delta greater than 0. but that doesn't make sense you need to define that relationship further. and if you define the relationship further like in the limit definition, you need to solve the proof for the statement to make sense.
     
  19. Jun 30, 2015 #18

    jbriggs444

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    The one is a formula with no free variables. The other is a formula with one free variable.

    Edit, I should be more clear here.

    It is common usage to interpret an if statement with one free variable as if it were implicitly qualified with a "for all" on the free variable. However, such a statement is then formally a "for all" rather than an "if then".
     
  20. Jun 30, 2015 #19

    Mark44

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    That's a very pedantic distinction.
     
  21. Jun 30, 2015 #20

    jbriggs444

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    Edit: Let me try to address this in more detail and in the process try to make it more clear why the distinction that Mark44 finds pedantic is one that I consider important.

    Yes. You can just say "x^2 > 0 where x > 0". Or "if x > 0 then x^2 > 0". The reader will normally assume that you mean to make an assertion about all possible x, not just an assertion about some specific x.

    To be pedantic, the logical formula: "if x > 0 then x^2 > 0" has one free variable, x. Its truth value could conceivably depend on the choice of x. By interpreting it as an assertion about all possible x, the reader is instead taking it as a shorthand way of expressing the logical formula: "for all x, if x > 0 then x^2 > 0". The latter formula has no free variables. It is either true (the if-then statement holds for all x) or it is false (the if-then statement fails to hold for at least one x).

    The short-hand rule of interpretation is handy. The writer has less to write. The reader has less to read. Everybody wins.

    But if one is nesting logical formulas inside of logical formulas, things can get difficult to interpret clearly. In the case of a definition of limits we have [at least] a "for all" enclosing a "there exists" which in turn encloses another "for all". In order to improve clarity, the "for all" statements are made explicitly. This way, the reader is not forced to assume anything to make sense out of the statement.

    Here is a formula with indentation to indicate the nesting.
    Code (Text):

    for all delta > 0
       there exists an epsilon > 0 such that
          for all x > epsilon
              |1/x - 0| < delta
     
    So for all epsilon > 0, there exists delta > 0 means for every number greater than 0 there exists a delta greater than 0.[/QUOTE]
    No. That is not correct. The statement goes on to make a further qualification on the relationship that some such delta has with the epsilon.
     
    Last edited: Jun 30, 2015
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