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Math Help - Continuity limit graph help

  1. #1
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    Exclamation Continuity limit graph help

    Please graph this piecewise function and answer the questions that follow.


    F(x)= { 1/x if x<1
    0 if x=1
    1 if x>1


    1.)Label any discontinuity and what type of discontinuity.
    2.)What values of c does - lim as x approaches c of F(x)- NOT exist?


    This is a question from my final test in my calc class. We don't get the test back .
    I don't get to hear from the instructor maybe a week from today. I'm SO eager to know the answers and if I did it right. Thanks in advance!
    Last edited by Mathinik; December 21st 2012 at 12:38 AM.
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    Re: Continuity limit graph help

    Hey Mathinik.

    You are definitely going to get a discontinuity at x = 1: can you guess why (Hint: continuity implies lim x->a f(x) = f(a): where does this condition fail?)
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    Re: Continuity limit graph help

    I answered, infinite discontinuity at x=0, jump discontinuity at x=1, and a removable discontinuity at x=1. Are those the right answers?

    Quote Originally Posted by chiro View Post
    Hey Mathinik.

    You are definitely going to get a discontinuity at x = 1: can you guess why (Hint: continuity implies lim x->a f(x) = f(a): where does this condition fail?)
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    Re: Continuity limit graph help

    I'm not sure what removable means but you are right about the other properties.
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    Re: Continuity limit graph help

    Removable or hole discontinuity
    Quote Originally Posted by chiro View Post
    I'm not sure what removable means but you are right about the other properties.
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    Re: Continuity limit graph help

    That sounds right then.
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    Re: Continuity limit graph help

    Need help still, from others?
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    Re: Continuity limit graph help

    Quote Originally Posted by Mathinik View Post
    I answered, infinite discontinuity at x=0, jump discontinuity at x=1, and a removable discontinuity at x=1. Are those the right answers?
    How can there be both jump discontinuity and removable discontinuity at x = 1? It should be removable. Concerning x = 0, Wikipedia stresses that a point of discontinuity must belong to the function's domain by definition, so x = 0 is not a point of discontinuity. This may not be a universal convention, so you need to check your sources.

    Quote Originally Posted by Mathinik View Post
    2.)What values of c does - lim as x approaches c of F(x)- NOT exist?
    c = 0.
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    Re: Continuity limit graph help

    Quote Originally Posted by Mathinik View Post
    Need help still, from others?
    Any other answer will depend upon which textbook you are following.
    The discontinuity at x=1 is most often called a removable discontinuity.

    The discontinuity at x=0 is called several different names.
    One being an essential discontinuity. I have seen authors who make a different name because there is no limit at x=0 as opposed to either \pm\infty being the limit. So, I were you, I would read the text.
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  10. #10
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    Re: Continuity limit graph help

    When I graph the function I also have the point (0,1) and together with 1/x from x<1 it has a jump discontinuity at x=1. X=0 is a point of discontinuity coz it has an asymptote/ break.

    C=O DNE, because the function will have an asymptote over there, so it is no continuous over there too. I answered c=0 and c=1(bec of the jump)



    Quote Originally Posted by emakarov View Post
    How can there be both jump discontinuity and removable discontinuity at x = 1? It should be removable. Concerning x = 0, Wikipedia stresses that a point of discontinuity must belong to the function's domain by definition, so x = 0 is not a point of discontinuity. This may not be a universal convention, so you need to check your sources.

    c = 0.
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    Re: Continuity limit graph help

    Are there two discontinuity at x=1? Based on the function's domain?

    Quote Originally Posted by Plato View Post
    Any other answer will depend upon which textbook you are following.
    The discontinuity at x=1 is most often called a removable discontinuity.

    The discontinuity at x=0 is called several different names.
    One being an essential discontinuity. I have seen authors who make a different name because there is no limit at x=0 as opposed to either \pm\infty being the limit. So, I were you, I would read the text.
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    Re: Continuity limit graph help

    Quote Originally Posted by Mathinik View Post
    Are there two discontinuity at x=1?

    No, there is only one. As I said, what it is called depends upon your textbook.
    Jump or removable are two impossibles.
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    Re: Continuity limit graph help

    Quote Originally Posted by emakarov View Post
    How can there be both jump discontinuity and removable discontinuity at x = 1? It should be removable. Concerning x = 0, Wikipedia stresses that a point of discontinuity must belong to the function's domain by definition, so x = 0 is not a point of discontinuity. This may not be a universal convention, so you need to check your sources.

    c = 0.
    Quote Originally Posted by Plato View Post
    No, there is only one. As I said, what it is called depends upon your textbook.
    Jump or removable are two impossibles.
    I answered wrong then, on my test.
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    Re: Continuity limit graph help

    Quote Originally Posted by Mathinik View Post
    When I graph the function I also have the point (0,1) and together with 1/x from x<1 it has a jump discontinuity at x=1.
    Do you mean you have (1,0) in the graph? The jump discontinuity is called this way not because there is a jump between the limit of the function (1 in this case) and the value of the function (0 in this case) at that point. The fact that there is a discontinuity already means that either the limit does not exist or it is not equal to the function value. Rather, it is called jump discontinuity because there are left and right limits and there is a jump between those. The value of the function does not matter. In this case, both left and right limits are 1, so this is a removable discontinuity (if f(1) were 1 instead of 0, the function would be continuous at x = 1).

    Quote Originally Posted by Mathinik View Post
    C=O DNE, because the function will have an asymptote over there, so it is no continuous over there too. I answered c=0 and c=1(bec of the jump)
    c = 0 is correct, but the limit when x -> 1 exists.

    Quote Originally Posted by Mathinik
    Are there two discontinuity at x=1?
    Quote Originally Posted by Plato
    No, there is only one. As I said, what it is called depends upon your textbook.
    Jump or removable are two impossibles.
    Do you mean, "two possibilities"?
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  15. #15
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    Re: Continuity limit graph help

    I was taught that you don't talk about a discontinuity in points were the function is not defined. As in this example the function is not defined in x = 0. Hence we don't have a discontinuity here.

    But at x = 1 we have a discontinuity. The book told us it was a removable discontinuity, and in this point we don't have lim{x->1} = f(1).

    For the second question we need to know that
    lim{x->a} f(x) = L <=> lim{x->a+} f(x) = lim{x->a-} f(x) = L
    to consider the function having a limit L at x = a.

    Thus we have no limit att x = 0 and neither at x = 1.
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