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Simplifying an Equation

  1. Oct 15, 2004 #1
    Technically this is a calculus problem I'm working on, but I'm just having problems with the Algebra portion.

    If I have:

    [tex](\frac{1}{x\sqrt{1+x}} - \frac{1}{x})[/tex]

    How can I simply this so that I can substitute in 0 for x?
     
  2. jcsd
  3. Oct 15, 2004 #2
    You want to get this in a form for the use of L'Hospital's Rule: [tex]\frac{1-\sqrt{1+x}}{x(\sqrt{1+x})}[/tex]

    In this form we see that as [tex]x\rightarrow0[/tex] the quotient is undefined, so we can differentiate and simplify.
     
    Last edited: Oct 15, 2004
  4. Oct 15, 2004 #3
    We haven't gone into differentiation or anything like that, is there another way?

    Actually, the problem that I'm trying to figure out is

    lim
    x -> 0 of the expression above.


    edit: For clarification - it's not for homework, it's just a problem I'm trying to figure out.
     
    Last edited: Oct 15, 2004
  5. Oct 15, 2004 #4
    I don't know any other way to do this problem. This is how you do it using the Calculus. You differentiate and get:

    [tex]\frac{[-2\sqrt{1+x}]^-1}{(2+3x)[2\sqrt{1+x}]^-1}=\frac{-1}{2+3x}\rightarrow \frac{-1}{2} ...as.... x \rightarrow 0[/tex]
     
    Last edited: Oct 15, 2004
  6. Oct 15, 2004 #5

    Fredrik

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    The substitution

    [tex]t=\sqrt{1+x}[/tex]

    simplifies the function to

    [tex]-\frac{1}{(1+t)t}[/tex]

    The limit of this as t goes to 1 is -1/2.
     
  7. Oct 15, 2004 #6
    That looks like a better way!
     
  8. Oct 15, 2004 #7

    shmoe

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    To add yet another way, rationalize the numerator. Multiply

    [tex]\frac{1-\sqrt{1+x}}{x(\sqrt{1+x})}[/tex]

    by

    [tex]\frac{1+\sqrt{1+x}}{1+\sqrt{1+x}}[/tex]

    to get

    [tex]\frac{-x}{x(\sqrt{1+x})(1+\sqrt{1+x})}[/tex]

    and go from there.
     
  9. Oct 15, 2004 #8

    Gokul43201

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    And finally see that #5 and #7 are really doing the same thing.

    They's both making use of the fact that x can be factored as [itex]-(1-\sqrt{1+x})(1+\sqrt{1+x}) [/itex]
     
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