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Math Help - conceptual question reguarding limits

  1. #1
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    conceptual question reguarding limits

    What is the difference between lim_{x\to\infty}\frac{a}{x} and \frac{x}{\infty}? When I first learned limits they were defined to me as when the independent variable of the function grows arbitrarily close to the value of the limit. Infinity was defined to me as when a number grows arbitrarily large. With this in mind, it makes very little sense to talk about a limit going to infinity.

    On a similar note, why is \frac{a}{0} indeterminant? Wouldn't it make more sense if it were \infty?
    Last edited by superdude; August 4th 2009 at 11:33 PM. Reason: fixed latex
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  2. #2
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    The first set of symbols you posted is read as "the limit as x increases without bound of a/x." In technical terms, it is defined as the number L such that for every positive difference e, there is some number N such that |L - a/x| < e for every x > N, if any such number L exists.
    Graphically, this is a horizontal asymptote, and in the function you gave, the horizontal asymptote is 0.
    The second symbol you posted is not defined as an operation in the real number system as \infty is not a real number. There are infinite numbers in other number systems, but you lose some of the algebraic structure of the real numbers, making some statements much more convoluted or simply not true.
    a/0, which is defined to be a\cdot 0^{-1} is not a real number, because 0 has no mutiplicative inverse. that is, there is no real number 0^{-1} such that 0\cdot 0^{-1} = 1.
    In limiting behavior, the limiting form a/0 is not indeterminate, it signifies that no limit exists. Note that when you write "infinite" limits, you are writing that the limit does not exist in a particular way. A limit is always a real number.
    In this case, consider \lim_{x\rightarrow 0} \frac{1}{x}. From the left, this limit increases without bound, while from the right this limit decreases without bound. Thus there is no limit or defined limiting behavior at 0, even though it has the form 1/0.
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  3. #3
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    Quote Originally Posted by superdude View Post
    What is the difference between lim_{x\to\infty}\frac{a}{x} and \frac{x}{\infty}? When I first learned limits they were defined to me as when the independent variable of the function grows arbitrarily close to the value of the limit. Infinity was defined to me as when a number grows arbitrarily large. With this in mind, it makes very little sense to talk about a limit going to infinity.
    "When a number grows arbitrarily large" is NOT a number! That's why \frac{a}{\infty} is not used- it makes no sense. Division is only defined for numbers and "infinity" is not a number.

    On a similar note, why is \frac{a}{0} indeterminant? Wouldn't it make more sense if it were \infty?
    Who told you it was indeterminant? If a\ne 0, it is not "indeterminant", it is "undefined"- which means literally that we can't define it: if we were to define \frac{a}{0}= X where X is a number then we would be saying a= 0(X) which is impossible because 0 times any number is 0. There is no such X. We don't say \infty because, again, \infty is not a number.

    "Indeterminant" is used for something like " \frac{0}{0}". We still cannot define it but for a different reason: if we say \frac{0}{0}= X, we are saying that 0= 0(X) and that is true for all numbers.
    Any number works for X and so that is "indeterminant". Do you see the difference?
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    allright, thanks
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  5. #5
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    one more thing

    why is it the definition of integrating has a limit going to infinity
    lim_{n\to\infty} \sum_{i=1}^n... where as power series have infinity \sum_{n=0}^\infty
    Last edited by superdude; August 8th 2009 at 10:50 PM. Reason: improved redability
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