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Simple Limits question regarding infinity

  1. Oct 7, 2012 #1
    1. The problem statement, all variables and given/known data


    I am doing some squeeze theorem questions, and I always run into things that is something divided by ∞, or something divided by √∞

    Why is it always zero?

    like.... -3/√∞ = 0 or -1/∞ = 0

    why??




    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 7, 2012 #2

    Zondrina

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    Homework Helper

    Think about this for a moment. Infinity is not an actual number we can calculate, rather a concept we use to determine what happens over a never ending period of time.

    For example consider :

    [tex] lim_{x→∞} \frac{1}{x} = 0 [/tex]

    So ask yourself for increasing values of x. Say x = 1, 2, 3, ......, n. f(x) is getting smaller and smaller and smaller the bigger and bigger the denominator gets. Notice the values of f(x) → 0 as x → ∞?

    So conceptually, some quantity over something really really reeeeally big tends to zero as the bigger thing gets bigger.

    Now what about the other case?

    [tex] lim_{x→0^+} \frac{1}{x} = ∞ [/tex]

    Same sort of argument here. Notice that for positive x = 1/2, 1/3, ..., 1/n, f(x) is getting bigger and bigger the smaller and smaller the denominator gets. So the values of f(x) → ∞ as x → 0.

    Does this help?
     
  4. Oct 7, 2012 #3

    HallsofIvy

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    If you "run into things that is something divided by [itex]\infty[/itex]", then you are doing the limits incorrectly- I suspect you are trying to "insert" [itex]\infty[/itex] into the formula and you can't do that- "infinity" is not a number.

    What you can say is that "limit as x (or n) goes to infinity" is shorthand for "as x (or n) gets larger and larger without bound:
    So [itex]\lim_{n\to\infty} \frac{-1}{n}[/itex] means we are looking at what happens as we make the denominator "larger, and larger, and larger, ....}. For example [itex]-1/1000000= -0.000001[/itex], [itex]-1/1000000000= -0.000000001, etc. Now, what do you think would happen if n got even larger and larger?

     
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