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Math Help - Continuity problem

  1. #1
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    Continuity problem

    Find constant A and B so that the given function will be continuous for all x values.

    Piece wise function f(x) = a(x)-4/x-2 if x does not equal 2, and b x = 2




    So confused lol
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  2. #2
    A Plied Mathematician
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    Be careful with your parentheses there. a x-(4/x)-2 is very different from a x-4/(x-2). Which one did you mean?

    Generally, I'd simply use the calculus definition of continuity here: make

    \displaystyle{\lim_{x\to 2}f(x)=f(2)}

    happen.
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  3. #3
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    It's the second one, ax-4/(x-2). I can't just substitute 2 in as it is is because it would give me 0 on the bottom so I don't know how to get rid of that on the bottom
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by Nikhiln25 View Post
    Find constant A and B so that the given function will be continuous for all x values.

    Piece wise function f(x) = a(x)-4/x-2 if x does not equal 2, and b x = 2




    So confused lol

    What is a(x)? Is it a*x? If yes, it should be ax^2 (I think..)
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  5. #5
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    Quote Originally Posted by Also sprach Zarathustra View Post
    What is a(x)? Is it a*x? If yes, it should be ax^2 (I think..)
    Yes a times x, not ax^2

    The problem in the book is given as

    ax-4
    -----
    x-2 if x=/= 2

    b if x=2
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Nikhiln25 View Post
    Yes a times x, not ax^2

    The problem in the book is given as

    ax-4
    -----
    x-2 if x=/= 2

    b if x=2
    You mean \displaystyle f(x) = \left \{ \begin{array}{ll} \frac {ax - 4}{x - 2} & \text{ if } x \ne 2 \\ & \\ b & \text{ if } x = 2  \end{array} \right. I take it.

    Note that you want \displaystyle ax - 4 to have a factor of \displaystyle x - 2 in it. So...?
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  7. #7
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    Quote Originally Posted by Nikhiln25 View Post
    It's the second one, ax-4/(x-2).
    Based on your subsequent post, this is incorrect. You mean (ax-4)/(x-2).

    Using universally accepted order of operations,

    ax-4/x-2 means \displaystyle ax - \frac{4}{x} - 2

    and

    ax-4/(x-2) means \displaystyle ax - \frac{4}{x-2}

    and

    (ax-4)/(x-2) means \displaystyle \frac{ax-4}{x-2}

    It wouldn't be a bad idea to learn how to use LaTeX either. This code

    [tex]f(x)=\begin{cases}\dfrac{ax-4}{x-2}&,\ x\ne2\\\\b&,\ x=2\end{cases}[/tex]

    produces

    f(x)=\begin{cases}\dfrac{ax-4}{x-2}&,\ x\ne2\\\\b&,\ x=2\end{cases}

    As to your question.. like Ackbeet said, you need \displaystyle \lim_{x \to 2} \frac{ax-4}{x-2}=b. In order to have this you will need to follow Jhevon's advice.
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  8. #8
    MHF Contributor Also sprach Zarathustra's Avatar
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    Hint:

    it's a=2


    Last edited by Also sprach Zarathustra; July 11th 2010 at 02:59 AM. Reason: not "useful" info.
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  9. #9
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    Hey, do like I do- erase the mistake so you can pretend you never made it!
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  10. #10
    MHF Contributor Also sprach Zarathustra's Avatar
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    Thank you for your advice!
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  11. #11
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    Thank you guys, using the hint also sparch jhevon posted I was able to get it.
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