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Math Help - Limits.. help needed!

  1. #1
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    Limits.. help needed!

    Hi,

    I have a problem in evaluating limits, especially when square roots are involved. I get so far and then hit a wall, although I know there is an answer.

    Could anyone please post a step by step breakdown to the following problem, so I can see at what stage I am going wrong.

    lim x-->2

    sqrt(5x-2) - 2sqrt(2)
    --------------------
    x^2 -3x +2

    Thanks in advance, I've been at this for hours!
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  2. #2
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    multiply numerator and denominator by ...

    \sqrt{5x-2} + 2\sqrt{2}
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  3. #3
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    reply

     \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2}  - 2\sqrt 2 }}<br />
{{x^2  - 3x + 2}} \hfill \\

    {\text{Rationalize the Numerator,}} \hfill \\

    \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2}  - 2\sqrt 2 }}<br />
{{x^2  - 3x + 2}} \times \frac{{\sqrt {5x - 2}  + 2\sqrt 2 }}<br />
{{\sqrt {5x - 2}  + 2\sqrt 2 }} \hfill \\


    Now, finish up.
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  4. #4
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    I've tried that, I get...

    (5x-2)-8
    -------------------------------
    (x^2-3x+2)(sqrt(5x-2)+2sqrt(2))

    when I plug 2 into x, it gives me

    0
    ---------
    (0)(sqrt(8)+2sqrt(2))

    which I make 0/0, but I know this cant be right. Any ideas?
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  5. #5
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    Quote Originally Posted by kwuk View Post
    I've tried that, I get...

    (5x-2)-8
    -------------------------------
    (x^2-3x+2)(sqrt(5x-2)+2sqrt(2))

    when I plug 2 into x, it gives me

    0
    ---------
    (0)(sqrt(8)+2sqrt(2))

    which I make 0/0, but I know this cant be right. Any ideas?
     \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2}  - 2\sqrt 2 }}<br />
{{x^2  - 3x + 2}} \hfill \\

    {\text{Rationalize the Numerator,}} \hfill \\

    \mathop { = \lim }\limits_{x \to 2} \frac{{\sqrt {5x - 2}  - 2\sqrt 2 }}<br />
{{x^2  - 3x + 2}} \times \frac{{\sqrt {5x - 2}  + 2\sqrt 2 }}<br />
{{\sqrt {5x - 2}  + 2\sqrt 2 }} \hfill \\

     \mathop { = \lim }\limits_{x \to 2} \frac{{(5x - 2) - 4(2)}}<br />
{{\left( {x^2  - 3x + 2} \right)\left( {\sqrt {5x - 2}  + 2\sqrt 2 } \right)}} \hfill \\

    \mathop { = \lim }\limits_{x \to 2} \frac{{5x - 10}}<br />
{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {\sqrt {5x - 2}  + 2\sqrt 2 } \right)}} \hfill \\

    \mathop { = \lim }\limits_{x \to 2} \frac{{5\left( {x - 2} \right)}}<br />
{{\left( {x - 1} \right)\left( {x - 2} \right)\left( {\sqrt {5x - 2}  + 2\sqrt 2 } \right)}} \hfill \\

    \mathop { = \lim }\limits_{x \to 2} \frac{5}<br />
{{\left( {x - 1} \right)\left( {\sqrt {5x - 2}  + 2\sqrt 2 } \right)}} \hfill \\

    Now, finish up. Did you get it Now???
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  6. #6
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    Thanks. Helped a lot. My factoring was all messed up.
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