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Limit problem simplify the root

  1. Jun 5, 2015 #1
    1. The problem statement, all variables and given/known data

    I feel like I'm missing some theorem which is preventing me from finalizing this problem! It's been driving me nuts I feel like I'm missing something super basic!

    Ultimately they've given the solution, g/8, so I know this is how I should try to get the equation to look algebraically. But, no matter how I manipulate it I cant get it to reduce.

    I feel like this has to do with √x2 = |x| Which then depending on your value of x will give x, or -x. However, I cant simplify the root into a way which will let me make this jump.
    upload_2015-6-5_12-25-51.png

    2. Relevant equations
    https://www.desmos.com/calculator/hf8poewlvb

    3. The attempt at a solution
    https://www.desmos.com/calculator/hf8poewlvb
    (link to all my reductions)
     
    Last edited: Jun 5, 2015
  2. jcsd
  3. Jun 5, 2015 #2

    LCKurtz

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    Try multiplying numerator and denominator by$$
    c\sqrt{\left(\frac{c^2}{g^2}+\frac 1 4\right)}+\frac{c^2}{g}$$
     
  4. Jun 5, 2015 #3
    The conjugate! How could I forget that.. Thank you! Now I have:

    PS. g is not squared I wrote it down wrong.
    VhM1czk.jpg

    Let me see what more I can do.
     
  5. Jun 5, 2015 #4

    pasmith

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    Alternatively: You can take a common factor out of the expression under the square root to obtain [tex]
    h(c) = A\sqrt{1 + x} - \frac{c^2}{g}[/tex] where [itex]x < 1[/itex] for sufficiently large [itex]c[/itex]. Hence you may expand the root as a binomial series, [tex]
    (1 + x)^{\alpha} = 1 + \alpha x + \frac{\alpha(\alpha - 1)}{2!}x^2 + \dots [/tex]
     
    Last edited: Jun 5, 2015
  6. Jun 5, 2015 #5
    Thanks for both of your help! Still working on it, sadly.. I pulled out a C so I'm left with:

    https://www.physicsforums.com/file:///C:/Users/Steven/Downloads/CodeCogsEqn.gif [Broken]
    7ScJ3py.gif

    Im not entirely sure what I could pull out of the root.
     
    Last edited by a moderator: May 7, 2017
  7. Jun 5, 2015 #6
    Factor out the c you'll have 1/4 * (lim c ->0 1/(sqrt(1/g^2 + 1/4c) + 1/g) and don't forget that 1/4c goes to 0 whenever c -> 0, good luck
     
    Last edited by a moderator: May 7, 2017
  8. Jun 5, 2015 #7

    SammyS

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    That's good.

    ##\displaystyle \ \frac14\lim_{c\to\infty}\left(\frac{c^2}{\displaystyle c\sqrt{\left(\frac{c^2}{g^2}+\frac 1 4\right)}+\frac{c^2}{g}}\right)\ ##

    One way to deal with rational expressions where some factor →∞ : divide the numerator and denominator by the highest power of that factor.

    Divide by c2 in the numerator & denominator.
     
  9. Jun 5, 2015 #8
    I got it! You guys rock thank you so much! Once I got down to that last x Term I realized that a constant/infinity = 0!!!

    Here is my work through!
    od4ZL7X.jpg
     
  10. Jun 5, 2015 #9
    You should drop the limit sign from the second last row.
     
  11. Jun 5, 2015 #10

    SammyS

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    At the point highlighted above, simply take the limit and simplify.
     
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