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Math Help - Please help for when a limit doesn't exist

  1. #1
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    Question Please help for when a limit doesn't exist

    Ok this is the problem:

    Find the limit as x approaches zero
    0^3 +x/x

    Ok.. so I know it's zero over zero so it's undefined but how can I further
    distribute the problem ? I remember my professor saying that we could take a step further but I don't know how to do it in this case.

    Hopefully this is enough information to get MUCH needed help
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  2. #2
    GAMMA Mathematics
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    Quote Originally Posted by kbgemini16 View Post
    Ok this is the problem:

    Find the limit as x approaches zero
    0^3 +x/x

    Ok.. so I know it's zero over zero so it's undefined but how can I further
    distribute the problem ? I remember my professor saying that we could take a step further but I don't know how to do it in this case.

    Hopefully this is enough information to get MUCH needed help
    What is your function?

    1) x^3 + \frac{x}{x}

    2) \frac{x^3 + x}{x}

    3) something else?
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  3. #3
    Junior Member teuthid's Avatar
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    L'Hopital

    This problem is a ripe candidate for L'Hopital's Law. Are you allowed to use it for this problem?
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  4. #4
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    When a limit doesn't exist

    it's #2 on your reply. And once I plugged the zero into the equation I got 0/0. In a similar problem done in class our professor further simplfied the answer by distributing the orignal function then cancelling out the like terms from the denominator. But I am somewhat fuzzy on the rest and don't know how to further simplify the answer.
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  5. #5
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    Quote Originally Posted by teuthid View Post
    This problem is a ripe candidate for L'Hopital's Law. Are you allowed to use it for this problem?

    Yes, I would assume so... He told us that 0/0 would not suffice as an answer. Please don't laugh but I've never even heard of that rule... I am researching it now
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  6. #6
    GAMMA Mathematics
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    Quote Originally Posted by kbgemini16 View Post
    it's #2 on your reply. And once I plugged the zero into the equation I got 0/0. In a similar problem done in class our professor further simplfied the answer by distributing the orignal function then cancelling out the like terms from the denominator. But I am somewhat fuzzy on the rest and don't know how to further simplify the answer.
    L'Hopital's rule requires you to know derivatives, which you may have not learned yet.

    Take a numerical approach to a limit when you are lost...

    f(x)=\frac{x^3+x}{x}

    f(1)=2

    f(0.5)=\frac{0.125+0.5}{0.5} \Rightarrow 1.25

    f(0.1)=\frac{0.001+0.1}{0.1} \Rightarrow \frac{0.101}{0.1} = 1.01

    f(0.01)=\frac{0.000001+0.01}{0.01} \Rightarrow \frac{0.010001}{0.01} = 1.0001

    It is obviously going to one.

    Now, think about it intuitively. What is happening? Very small values that are cubed are comparable to zero. What is left? Just x/x which simplifies to one.

    Also, note that you can just simplify the function...

    f(x)=\frac{x^3+x}{x} \Rightarrow x^2+1 for x \ne 0

    Limits do not take the value of a function at the limiting value, but merely the value an infinitesimally small distance away from the limiting value, so doing this primitive simplification is allowable since the limit never looks at f(0) which so happens to be undefined.
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  7. #7
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    Quote Originally Posted by teuthid View Post
    This problem is a ripe candidate for L'Hopital's Law. Are you allowed to use it for this problem?
    Sorry, but l'Hospital's Rule is the lazy way.

    Quote Originally Posted by colby2152 View Post
    [snip]
    you can just simplify the function...

    f(x)=\frac{x^3+x}{x} \Rightarrow x^2+1 for x \ne 0

    Limits do not take the value of a function at the limiting value, but merely the value an infinitesimally small distance away from the limiting value, so doing this primitive simplification is allowable since the limit never looks at f(0) which so happens to be undefined.
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