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Thread: Limits and Infinity

  1. #1 Limits and Infinity 
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    The statement " lim as n tends to infinity of 1/n = 0 is false! The understanding requires a conceptual jump in logic from "tend to" to "reaced".

    It is not enough to tend to infinity since then the computation does not give exactly zero. One must actually reach infinity and plug it into n to get exactly zero.

    We cannot reach infinity practically so it must be proved that we can reach it theoreticly. It is my contention that we cannot even reach it theoreticly and this is implicit in the mathematical use of lim instead of f(infinity).

    Saying that we can get as close to zero as we wish by making n large enough does not bridge the logical gap between the concepts "tend to" and "reached" (where these concepts relate to numbers).


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  3. #2  
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    The statement " lim as n tends to infinity of 1/n = 0 is false!
    In maths everything is a matter of definitions and conventions.

    The mathematical meaning of your statement, is, using commonly accepted definitions and conventions :

    Given any number it exists a number such as for any x such as
    then . And this statement is true.

    the lim statement, as you formulated it, is a convenient shortcut, a writing convention, but its exact mathematical meaning is the one above.

    It is important to note that "infinity" is NOT a number in the real numbers set : is meaningless and must not be written for real numbers. Infinity just means "as large as you want".

    Does it make things clearer ?


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  4. #3  
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    Quote Originally Posted by talanum1 View Post
    One must actually reach infinity and plug it into n to get exactly zero.
    We're not plugging into a function, however. But we are plugging in , for the function .

    When we "plug in" zero for , the result is . This is undefined. But that's completely irrelevant.

    The entire point of a limit is not to see what the actual value is, but to see where it looks like the function is going.

    And for , as gets closer and closer to from the right side, grows indefinitely (=towards infinity).
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    Have you studied the definition of a limit? Use it to prove

    It seems that you haven't really studied limits, I suggest you do your own homework before declaring something is true or false.
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  6. #5  
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    As for me i have always been confused between infinity as a number and infinity as a concept.
     

  7. #6  
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    Infinity isn't a number.
     

  8. #7  
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    Beware: The subject of infinity can lead to insanity! My experience in world mathematic realtes to finite concepts: amounts that you can count and express in symbols. The only concepts that I tend to believe are consistent with the concept of "infinity" is space and objects in motion in the form of revolutions. Unless there is an exception, accountability of matter via mathematics should be finite in nature. In other words I do not think I can divide and subdivide a pizza into an infinite number of parts (fractions of the whole).

    In the case of dividing a pizza into parts, we can express the division mathematically in the following form:
    1.0 W/n, where W is 100% as a whole and n (the amount of parts the pizza is divided into) is any whole positive number, except 0. Note, the division by n causes the following effect: as n increases, the actual real size of each part (unit fraction of the whole) decreases...toward zero size! Or should I say...toward a miniature infinity!
    And... by the same token, but in the "opposite direction," as the n (number of parts in the pizza) decreases, the size of each part (unit fraction in reality) increases toward the other kind of infinity: "toward a greater infinity!"
    I wonder what would happen if I were to substitute the concept of space in place of the pizza and divide it into parts?
    eaglepass ...My quote: "In order to think out of the box, you must first get out of the box."
    Last edited by eaglepass; January 15th, 2013 at 10:38 PM. Reason: correcting grammar
     

  9. #8  
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    Quote Originally Posted by eaglepass View Post
    What I have experience in world mathematic is finite concepts: amounts that you can count
    Then, if I may say so, your experience of mathematics is not very extensive.

    A set is said to be "countable" if it can be put into one-to-one correspondence with a subset of , the counting numbers. Since, by a definition of set theory, any set is a subset of itself, and since the cardinality of is infinite, then a countable set may be infinite. It is called a "countably infinite set", it being understood that this refers to its cardinality
     

  10. #9  
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    I never could grasp the concept of infinity, with my limited understanding of mathematics. If it cannot be divided, it must be 0, but infinity also cannot be divided. I'm deeply confused.
     

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    Quote Originally Posted by Guitarist View Post
    Quote Originally Posted by eaglepass View Post
    What I have experience in world mathematic is finite concepts: amounts that you can count
    Then, if I may say so, your experience of mathematics is not very extensive.

    A set is said to be "countable" if it can be put into one-to-one correspondence with a subset of , the counting numbers. Since, by a definition of set theory, any set is a subset of itself, and since the cardinality of is infinite, then a countable set may be infinite. It is called a "countably infinite set", it being understood that this refers to its cardinality
    It is my understanding that a countable set "may be" infinite. Can you explain what "may be infinite" means. Does this mean that a countable set may also be finite if the amount of elements in it are less than N? It appears that when we use the counting numbers, we use them to establish an amount of items in a whole or any container. If a pizza is divided into 8 equal parts, we can use the counting numbers to describe that particular amount of items: 1/8+1/8/+1/8+1/8+1/8+1/8+1/8+1/8. We start by using 1,2,3,4,5,6,7,8 in a one-to-one correspondence forming an ordinal sequence. The natural number 8 is a member of the infinite set N and is less than N. Thus the number 8 is a member of a finite set that can be counted via the counting numbers, not the natural numbers which are considered to be infinite. My quote: "In order to think out of the box, you must first step out of the box." eaglepass
     

  12. #11  
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    Quote Originally Posted by eaglepass View Post

    It is my understanding that a countable set "may be" infinite. Can you explain what "may be infinite" means. Does this mean that a countable set may also be finite if the amount of elements in it are less than N?
    Yes, quite obviously
    The natural number 8 is a member of the infinite set N and is less than N. Thus the number 8 is a member of a finite set that can be counted via the counting numbers, not the natural numbers which are considered to be infinite.
    This makes no sense, I am afraid. Check your understanding of the difference between ordinal and cardinal numbers. Check also the fact that any set of whatever cardinality (countable or uncountable,say) contains as a subset a countable set, finite or otherwise
    My quote: "In order to think out of the box, you must first step out of the box."
    My quote: "In order to think out of the box, you first have to learn how to think"
     

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    I tend to agree with you. In my opinion, symbols, definitions, and conventions tend to trap us in a "mental box" as "they" lead us from reality to fantasy via mathematical illusions. Then some Mr. Perfect and/or Mr. Know It All will stifle the forward progress in knowledge by using definitions and conventions to keep others from trying to "think out side the box." If I may, I would like to step outside the box and do some "thinking" related to your comments by making a reference to the concept of the WHOLE and ITS PARTS.
    If a pizza is divided into two equal pieces, then each piece may be considered to be a unit fraction of the whole. This can be expressed in terms of repeated addition: 1/2+1/2= 1.0 W, where W represents the whole pizza. And from elementary algebra, we can show that 1/2+1/2= 2(1/2)= 2/2=1.0.
    If the pizza is divided into three equal parts, you are going to note that the size of each piece decreases, and the amount of pieces increases, even though each piece is smaller. We should also note that the size of the pizza remains the same size: it does not increase or decrease! This actual division may also be expressed in terms of repeated addition of unit fractions to form the whole: 1/3+1/3+1/3=1.0 W.
    What if the pizza is divided into ten parts? The unit fraction of the whole in this case is 1/10 W. In decimal terms, it is expressed as
    0.1 W. In terms of percent, it is expressed as 10% of W. Again, note that the actual size of each unit fraction of the whole decreases as the amount of pieces of pizza increases. Also note that the size of the pizza remains the same.
    As you can see, and I hope you agree with me, that the value of the unit fraction is a "function" of the amount of equal parts the pizza (1.0 W) is in reality divided into. If the pizza is divided into x amount of equal parts, then each equal part is considered to be a unit fraction of the whole. Thus the value of each unit fraction may be expressed as (1/x) (1.0 W).
    Note: we can prove that x(1/x 1.0 W)=x/x 1.0 W=1.0 W! Thus I tend to agree that the statement you referred to in you comment: " lim as n tends to infinity of 1/n = 0 is false! appears to be true since the measure of the unit fraction 1/n is a function of the amount of parts the whole is divided into, and the other way around. In this case, the [So this is where we get x(1/x) = 1 from!] value of "N" does not appear to depend on the value of"1/n" since the whole, in this case, the pizza, remains the same: the pizza does not increase or decrease! If I may, I would like to use "symbols and convention" to express the relationship between "N" and 1/n: N=>1/n. This expression is an example of out-of-box-thinking: N is the amount of equal parts the whole is divided into; 1/n represents the size of the unit fraction. LOOK=>: N is presumed to be a real amount; BUT 1/n relates to size!
    I consider this approach toward infinity: "lim as n tends to infinity of 1/n = 0," to be the dividing line between reality and fantasy. This is where I believe "infinity" becomes an illusion similar to the one Albert Einstein used in the theory of relativity. MY QUOTE: IN ORDER TO THINK OUT OF THE BOX, ONE MUST FIRST STEP OUT OF THE BOX!
    Last edited by eaglepass; January 17th, 2013 at 12:58 AM.
     

  14. #13  
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    Fortunately for the rest of us, mathematics is a bit more interesting that a virtual Pizza Hut.
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    Guitarist



    In your statement/comment, of January 17, 1013, "Fortunately for the rest of us, mathematics is a bit more interesting that a virtual Pizza Hut," you are including "the rest of us" without proof. Your statement is your own opinion. What is "interesting" to "the rest of us" is a matter of personal choice by "the rest of us," and not by you.

    In my opinion
    , a person's interest is in the mind, heart, and soul of each beholder. In my opinion, some of your comments I have read throughout this forum, reflect an air of arrogance. It would be much in the interest of the progress of this forum, which I like very much because I learn from the threads and replies by the participants, for you to consider behaving more like a professional "user friendly" moderator than an ARROGANT BULLY!

    MY QUOTE: IN ORDER TO THINK OUT OF THE BOX, ONE MUST FIRST STEP OUT OF THE BOX!
    eaglepass
     

  16. #15  
    Brassica oleracea Strange's Avatar
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    Quote Originally Posted by eaglepass View Post
    In your statement/comment, of January 17, 1013, "Fortunately for the rest of us, mathematics is a bit more interesting that a virtual Pizza Hut," you are including "the rest of us" without proof.
    I can't imagine anyone being interested in your bizarre and incoherent ramblings about pizzas. Maybe you should join a foodie forum; it doesn't seem to have any place in a math forum. (Maybe you need to find a bigger box to "think outside of".)
    Without wishing to overstate my case, everything in the observable universe definitely has its origins in Northamptonshire -- Alan Moore
     

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    TO STRANGE...IN MY OPINION, YOU ARE STILL IN THE BOX AND POSSIBLY CAN'T PERCEIVE IT! THAT IS WHY YOU CAN'T IMAGINE ANYONE BEING INTERESTED IN MY BIZZARE AND INCOHERENT RAMBLINGS ABOUT PIZZAS. IT TAKES SOMEONE WITH THE MENTAL SKILLS TO READ WHAT I WROTE ABOUT PIZZAS AND THE MATH RELATED TO IT. THE FACT THAT YOU CAN'T IMAGINE....MEANS YOU HAVE THE SAME TYPICAL PROBLEM MOST PEOPLE HAVE ONCE THEY ARE STIFFLED BY THE RULES OF MATHEMATICS.....GIVE YOURSELF A CHANCE TO EXAMINE WHY I RAMBLE AND WHY YOU THINK I RAMBLE. YOU MAY LEARN SOMETHING....OR STAY STIFFLED IN THE BOX! eaglepass
     

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    Sorry chum, thread locked. Reason...... Stupidity
     

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