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Math Help - "Theory" on Indeterminate forms

  1. #1
    s3a
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    "Theory" on Indeterminate forms

    Question:
    For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value, or never has a fixed finite value. In the first case answer IND, in the second case enter the numerical value, and in the third case answer DNE. For example:
    For you would answer IND,
    for you would answer 0,
    and for you would answer DNE.
    NOTE: If the answer is or , answer DNE, since infinite limits cannot strictly be said to exist.
    Note that l'Hopital's rule (in some form) may ONLY be applied to indeterminate forms.

    1.

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    5.

    6.

    7.

    8.

    9.

    10.

    11.

    12.

    13.

    14.

    15.

    16.

    17.

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    19.

    20.

    I apparently got 17 out of 20 of these right so which of these two did I get wrong and why? Also, I copied 1^infinity as being indeterminate from my book (got it right above) but would otherwise have guessed that 1^infinity = inifinity = DNE (in the context of this question) so could someone explain to me why it is wrong? My logic is that for example 1.00000000000001^huge number = inifnity. Another example try doing 1.01^2 and then 1.01^3 and so on and you will see that it keeps growing.

    Any help would be greatly appreciated!
    Thanks in advance!
    Last edited by s3a; December 1st 2009 at 05:11 AM.
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  2. #2
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    Quote Originally Posted by s3a View Post
    Question:
    For each of the following forms determine whether the following limit type is indeterminate, always has a fixed finite value, or never has a fixed finite value. In the first case answer IND, in the second case enter the numerical value, and in the third case answer DNE. For example:
    For you would answer IND,
    for you would answer 0,
    and for you would answer DNE.
    NOTE: If the answer is or , answer DNE, since infinite limits cannot strictly be said to exist.
    Note that l'Hopital's rule (in some form) may ONLY be applied to indeterminate forms.

    1.

    2.

    3.

    4.

    5.

    6.

    7.

    8.

    9.

    10.

    11.

    12.

    13.

    14.

    15.

    16.

    17.

    18.

    19.

    20.

    I apparently got 17 out of 19 of these right so which of these two did I get wrong and why? Also, I copied 1^infinity as being indeterminate from my book (got it right above) but would otherwise have guessed that 1^infinity = inifinity = DNE (in the context of this question) so could someone explain to me why it is wrong? My logic is that for example 1.00000000000001^huge number = inifnity. Another example try doing 1.01^2 and then 1.01^3 and so on and you will see that it keeps growing.

    Any help would be greatly appreciated!
    Thanks in advance!
    Please give your answer to each one in order to facilitate assistance.

    As for your logic that 1^{\infty} = \infty, surely you have met \lim_{n \to + \infty} \left(1 + \frac{1}{n} \right)^n = e ....
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  3. #3
    s3a
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    Oh, sorry I forgot to preview again. Here is an image file (.png) that has the questions and my answers. (Apparently I have 17/20 right) And so (~1)^infinity = e and e is somewhat like infinity since the number never ends but it's not because it's just a number, right (irrational number)?
    Attached Thumbnails Attached Thumbnails "Theory" on Indeterminate forms-answers-i-put.png  
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  4. #4
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    Well I just took a quick glance but, nr 7 is indeterminate.

    For example consider:

    \lim_{x\to\infty}\sqrt{x}-\sqrt{x-1} now this surely has the form \infty-\infty

    But

    =\lim_{x\to\infty}\left(\sqrt{x}-\sqrt{x-1}\right)\cdot\frac{\sqrt{x}+\sqrt{x-1}}{\sqrt{x}+\sqrt{x-1}}=\lim_{x\to\infty}\frac{x-(x-1)}{\sqrt{x}+\sqrt{x-1}}=\lim_{x\to\infty}\frac{1}{\sqrt{x}+\sqrt{x-1}}=0
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  5. #5
    s3a
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    Thanks, I actually knew that one though and just missed it. What about my other mistakes?
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  6. #6
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    Quote Originally Posted by s3a View Post
    Thanks, I actually knew that one though and just missed it. What about my other mistakes?
    I have time for one:

    #5 is wrong. It is an indeterminant form. eg. - Wolfram|Alpha
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