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Math Help - Calc hmwk- extend the fucntion and remove discontinuity

  1. #1
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    Calc hmwk- extend the fucntion and remove discontinuity

    I don't understand how to remove the discontinuity on this problem:

    (x-4)/( (sq. root of x) - 2), x=4

    please help!
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  2. #2
    Moo
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    Hello,
    Quote Originally Posted by h4hv4hd4si4n View Post
    I don't understand how to remove the discontinuity on this problem:

    (x-4)/( (sq. root of x) - 2), x=4

    please help!
    Let f(x)=\frac{x-4}{\sqrt{x}-2}

    There's clearly a discontinuity at x=4.

    Now, if you find out that \lim_{x \to 4^+} f(x)=\lim_{x \to 4^-} f(x)=L, then you can define the function g :

    g(x)=\left\{\begin{array}{ll} f(x) \quad \forall x \neq 4 \\ L \quad \text{for } x=4 \end{array} \right.

    This removed the discontinuity
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  3. #3
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    I'm sorry, I don't think I asked the right question, here goes: is x=4 the extended function?
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  4. #4
    Moo
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    Quote Originally Posted by h4hv4hd4si4n View Post
    I'm sorry, I don't think I asked the right question, here goes: is x=4 the extended function?
    If I understood well, the extended function is g.
    The domain of f was any real number except 4.
    The domain of g is any real number.
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  5. #5
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    Hello, h4hv4hd4si4n!

    Remove the discontinuity: . f(x) \;=\;\frac{x-4}{\sqrt{x} - 2}

    Since f(4) \:=\:\frac{4-4}{\sqrt{4}-2} \:=\:\frac{0}{0}, there is a discontinuity at x = 4.
    Code:
            |
            |                 *
            |          *
            |      o
            |   *  :
            | *    :
            |*     :
            |      :
           2*      :
            |      :
            |      :
        ----+------+-------------
            |      4
    It is the upper half of a "horizontal" parabola
    . . with a hole at (4, 4).


    Rationalize: . \frac{x-4}{\sqrt{x}-2}\cdot\frac{\sqrt{x} + 2}{\sqrt{x}+2} \;=\;\frac{(x-4)(\sqrt{x} + 2)}{x - 4} \;=\;\sqrt{x} + 2

    Hence: . \lim_{x\to4}f(x) \;=\;\lim_{x\to4}(\sqrt{x}+2) \;=\;4


    Therefore: . f(x) \;=\;\bigg\{\begin{array}{ccc}\frac{x-4}{\sqrt{x}-2} & & x \neq 4 \\ 4 & & x = 4 \end{array}

    . . and this "fills in the hole" . . . The function is continuous.

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