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Math Help - Evaluate the limits

  1. #1
    Member dokrbb's Avatar
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    Evaluate the limits

    I have such a problem:

    Given that lim x->a f(x) = 0, lim x->a g(x) = 0, lim x->a h(x) = 1, lim x->a p(x) = INF, lim x->a q(x) = INF,

    Evaluate these limits:

    1. Lim x->a f(x)/g(x) =
    2. Lim x->a f(x)/p(x) =
    3. Lim x->a h(x)/p(x) =
    4. Lim x->a p(x)/f(x) =
    5. Lim x->a p(x)/q(x) =

    I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we donít have enough info to determine the limit.

    1. Lim x->a f(x)/g(x) = 0/0, (DNE)
    2. Lim x->a f(x)/p(x) = 0/INF, (Indeterminate form)
    3. Lim x->a h(x)/p(x) = 1/INF, (positive infinitive)
    4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
    5. Lim x->a p(x)/q(x) = INF/INF, (DNE)

    I'm not quite sure in my reasoning so I need your input, thanks
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    Re: Evaluate the limits

    Quote Originally Posted by dokrbb View Post
    I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we donít have enough info to determine the limit.
    Could you remind what an indeterminate form is?

    Quote Originally Posted by dokrbb View Post
    Lim x->a f(x)/g(x) = 0/0, (DNE)
    And what if f(x) = g(x)?

    Quote Originally Posted by dokrbb View Post
    Lim x->a f(x)/p(x) = 0/INF, (Indeterminate form)
    And what if a = 0, f(x) = x and p(x) = 1/x?

    Quote Originally Posted by dokrbb View Post
    Lim x->a h(x)/p(x) = 1/INF, (positive infinitive)
    The bigger the denominator, the bigger the ratio?

    Quote Originally Posted by dokrbb View Post
    Lim x->a p(x)/q(x) = INF/INF, (DNE)
    And what if p(x) = q(x)?
    Thanks from dokrbb
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  3. #3
    Member dokrbb's Avatar
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    Re: Evaluate the limits

    Quote Originally Posted by emakarov View Post
    Could you remind what an indeterminate form is?
    it is a result that doesn't provide enough info about the actual limit after replacing the values in the expression,

    so, wait a min, if by indeterminate forms is meant 0, 0/0, inf, inf/inf, it means I have some more indeterminate forms in my problem?

    Quote Originally Posted by emakarov View Post

    And what if f(x) = g(x)
    you mean what if 0 = 0, ? but this is also an indeterminate form, isn't it?


    Quote Originally Posted by emakarov View Post

    And what if a = 0, f(x) = x and p(x) = 1/x?
    then f(x)=0 and p(x)=1/0 and are indeterminate forms

    Quote Originally Posted by emakarov View Post

    The bigger the denominator, the bigger the ratio?
    well, actually, the bigger the denominator, more the values of f(x) approaches 0 (from how I understood, lesser and lesser parts of 1 we obtain, e.g. 0,5; 0,001; 0,00001 and so on)


    Quote Originally Posted by emakarov View Post
    And what if p(x) = q(x)?
    but this means that INF = INF
    => indeterminate forms, too?
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    Re: Evaluate the limits

    Quote Originally Posted by dokrbb View Post
    Given that lim x->a f(x) = 0, lim x->a g(x) = 0, lim x->a h(x) = 1, lim x->a p(x) = INF, lim x->a q(x) = INF,
    Evaluate these limits:
    1. Lim x->a f(x)/g(x) =
    2. Lim x->a f(x)/p(x) =
    3. Lim x->a h(x)/p(x) =
    4. Lim x->a p(x)/f(x) =
    5. Lim x->a p(x)/q(x) =

    I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we donít have enough info to determine the limit.
    1. Lim x->a f(x)/g(x) = 0/0, (DNE)
    2. Lim x->a f(x)/p(x) = 0/INF, (Indeterminate form)
    3. Lim x->a h(x)/p(x) = 1/INF, (positive infinitive)
    4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
    5. Lim x->a p(x)/q(x) = INF/INF, (DNE)
    Look I know that I am not as nice an instructor as emakarov.
    But I must tell you that I think that you have all of these concepts confused.

    Lets look at #2,
    You have a fraction. Its numerator is very 'close' to 0.
    Its denominator is very large in a positive sense.

    Here is an example: ~\frac{10^{-100}}{10^{100}}.
    Of what do you think that fraction is approximation ?

    Now you have correctly listed Indeterminate forms, of which #2 is none of.

    So can you repost this so that it makes sense?
    Thanks from dokrbb
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    Re: Evaluate the limits

    Quote Originally Posted by Plato View Post
    Look I know that I am not as nice an instructor as emakarov.
    But I must tell you that I think that you have all of these concepts confused.

    Lets look at #2,
    You have a fraction. Its numerator is very 'close' to 0.
    Its denominator is very large in a positive sense.

    Here is an example: ~\frac{10^{-100}}{10^{100}}.
    Of what do you think that fraction is approximation ?

    Now you have correctly listed Indeterminate forms, of which #2 is none of.

    So can you repost this so that it makes sense?
    If I were to have such an example I would write it as [1/10^100]/[10^100] = [1/10^100]*[1/10^100]... so, the limit DNE

    if the lim x->a f(x) approaches o and lim x->a p(x) approaches INF =>the lim DNE

    1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
    2. Lim x->a f(x)/p(x) = 0/INF, (DNE)
    3. Lim x->a h(x)/p(x) = 1/INF, (Indeterminate form)
    4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
    5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)

    am I right (bc I feel I'm going to be more confused... )

    ps: don't worry about the niceness while giving me advices, I appreciate much more the advices, not the way they are given
    Last edited by dokrbb; March 30th 2013 at 06:51 PM.
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    Re: Evaluate the limits

    wrong post, sorry
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    Re: Evaluate the limits

    Quote Originally Posted by dokrbb View Post
    If I were to have such an example I would write it as [1/10^100]/[10^100] = [1/10^100]*[1/10^100]... so, the limit DNE

    if the lim x->a f(x) approaches o and lim x->a p(x) approaches INF =>the lim DNE

    1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
    2. Lim x->a f(x)/p(x) = 0/INF, (DNE)
    3. Lim x->a h(x)/p(x) = 1/INF, (Indeterminate form)
    4. Lim x->a p(x)/f(x) = INF/0, (Indeterminate form)
    5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)

    am I right (bc I feel I'm going to be more confused... )
    so...?
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    Re: Evaluate the limits

    Quote Originally Posted by dokrbb View Post
    so...?
    It truly troubles me that you are seemingly in a calculus course but know so very little about limits.

    Please study this page on indeterminate forms.
    LEARN all of those forms. Then complete rework what you posted, because most of your answers are completely wrong.
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    Re: Evaluate the limits

    Thanks Plato, I got them:

    1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
    2. Lim x->a f(x)/p(x) = 0/INF = 0 bc as f(x) approaches 0, p(x) becomes large
    3. Lim x->a h(x)/p(x) = 1/INF = 0 since when h(x) approaches a finite nr p(x) becomes large
    4. Lim x->a p(x)/f(x) = INF/0, DNE (I think we would need to evaluate limit from the right and from the left in order to be able to determine the lim)
    5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)
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    Re: Evaluate the limits

    Quote Originally Posted by dokrbb View Post
    I got them:
    1. Lim x->a f(x)/g(x) = 0/0, (Indeterminate form)
    2. Lim x->a f(x)/p(x) = 0/INF = 0 bc as f(x) approaches 0, p(x) becomes large
    3. Lim x->a h(x)/p(x) = 1/INF = 0 since when h(x) approaches a finite nr p(x) becomes large
    4. Lim x->a p(x)/f(x) = INF/0, DNE (I think we would need to evaluate limit from the right and from the left in order to be able to determine the lim)
    5. Lim x->a p(x)/q(x) = INF/INF, (Indeterminate form)
    And that looks much better.
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    Re: Evaluate the limits

    The only remark is the following.

    Quote Originally Posted by dokrbb View Post
    4. Lim x->a p(x)/f(x) = INF/0, DNE (I think we would need to evaluate limit from the right and from the left in order to be able to determine the lim)
    I am not sure if INF is included in DNE or not. The limit \lim_{x\to a}p(x)/f(x) can be \pm\infty or it may not exist. The former happens when f(x) does not change sign in some neighborhood of a, and the latter happens when there are both positive and negative values f(x) when x is arbitrarily close to a: then p(x) / f(x) changes sign as x tends to a while its absolute value continues to grow.
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    Re: Evaluate the limits

    Quote Originally Posted by emakarov View Post
    The only remark is the following.

    I am not sure if INF is included in DNE or not. The limit \lim_{x\to a}p(x)/f(x) can be \pm\infty or it may not exist. The former happens when f(x) does not change sign in some neighborhood of a, and the latter happens when there are both positive and negative values f(x) when x is arbitrarily close to a: then p(x) / f(x) changes sign as x tends to a while its absolute value continues to grow.
    well, yes but since I'm allowed to give separate answers: whether \lim_{x\to a}p(x)/f(x) is a) \p\infty ,b) -\infty , c) DNE or don't have enough information to evaluate it, I considered the more appropriate answer in this case is c)

    and thanks a lot
    Last edited by dokrbb; March 31st 2013 at 11:57 AM.
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    Re: Evaluate the limits

    Quote Originally Posted by dokrbb View Post
    well, yes but since I'm allowed to give separate answers: whether \lim_{x\to a}p(x)/f(x) is a) \infty ,b) -\infty , c) DNE or don't have enough information to evaluate it, I considered the more appropriate answer in this case is c)
    When you say that the answer is c), it means that the limit does not exists for all function p(x) and f(x) under given assumptions. And this is not true: for some p(x) and f(x), the limit is +∞ or -∞. Therefore, I think the correct answer is that there is not enough information.
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    Re: Evaluate the limits

    Quote Originally Posted by emakarov View Post
    When you say that the answer is c), it means that the limit does not exists for all function p(x) and f(x) under given assumptions. And this is not true: for some p(x) and f(x), the limit is +∞ or -∞. Therefore, I think the correct answer is that there is not enough information.
    Frankly I am confused at this point, myself.
    In the OP we are given that {\lim _{x \to a}}f(x) = 0\quad \& \quad {\lim _{x \to a}}p(x) = \infty .

    We are also told that "c) DNE or don't have enough information to evaluate it" Also "I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we donít have enough info to determine the limit[I]."
    I did not think that "that there is not enough information" was an option.
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    Re: Evaluate the limits

    Quote Originally Posted by Plato View Post
    Frankly I am confused at this point, myself.
    In the OP we are given that {\lim _{x \to a}}f(x) = 0\quad \& \quad {\lim _{x \to a}}p(x) = \infty .

    We are also told that "c) DNE or don't have enough information to evaluate it" Also "I have to mention which of these limits are indeterminate forms, positive INF, Negative INF, or the limit does not exist (DNE) or we donít have enough info to determine the limit[I]."
    I did not think that "that there is not enough information" was an option.
    the confusing statement is that for c) the both answers are mentioned (1)DNE or 2) not enough info); therefore, whether I consider the situation 1) or 2), I have to chose the same c) answer,

    sorry, but that's how the options were stated, and I checked the answers - it was c) the correct answer
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