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Homework Help: Solving this limit

  1. Nov 16, 2014 #1
    1. The problem statement, all variables and given/known data

    lim x→2
    (4-(√18-x)) / (x-2)

    Note that the square root goes over the 18 AND the x, not just the 18.

    I don't know how to use the fancy mathematical notation on this forum and I have no idea where to go to find out how to use it.

    2. Relevant equations
    There are absolutely NO relevant equations. ZERO!

    3. The attempt at a solution
    I multiplied the numerator and the denominator by the conjugate of the numerator, because direct substitution would yield an indeterminate. However, after distributing, direct substitution of 2 still gave me an indeterminate.

    Here is my distribution
    (16-(18-x)) / (4x-x(√18-x)-8+(√18-x))

    How can I further simplify this?
  2. jcsd
  3. Nov 16, 2014 #2


    User Avatar
    Gold Member

    Have you learned about L'Hoptial's rule yet?
  4. Nov 16, 2014 #3
    Don't FOIL the denominator. Simplify the numerator. You should see a fairly easy cancellation.
  5. Nov 16, 2014 #4
    No, I haven't!

    I think I see what you mean, but doesn't 16-(18-x) simplify to (-2 - x)? My denominator has (x-2) in it.
  6. Nov 16, 2014 #5
    You didn't distribute the minus sign.
  7. Nov 16, 2014 #6


    Staff: Mentor

    For problems like this, L'Hopital's rule is often ineffective. The approach taken by the OP is the better way to go here.
  8. Nov 16, 2014 #7
    DUH. Can't believe I forgot to do that. Makes much more sense now.

    However, after I simplify and cancel, I'm left with a denominator that = 0 when x = 2.

    -1 / (4 - (√18 - x))

    After I graphed it, I found there's no limit. (Approaching positive and negative infinity) However, how do I algebraically prove that there is no limit in a function? Sorry for all of these questions - I'm studying limits on my own so I might be missing certain important concepts here and there.
  9. Nov 16, 2014 #8


    Staff: Mentor

    I think you made a mistake. You should end up with 4 + √(18 - x) in the denominator.

    Note the parentheses I used to indicate that what's under the radical is 18 - x, not just 18.
  10. Nov 22, 2014 #9
    I would start with making a replacement [itex]y=x-2[/itex]. You can further simplify this by taking [itex]z=\frac{y}{16}[/itex]. After that just follow the way you did before, but the expression should look much more pleasant now.
  11. Nov 22, 2014 #10
    Multiply the numerator and denominator by the conjugate of ##4 - \sqrt{18-x}##. From there, it should work out if you simplify everything right.
  12. Nov 22, 2014 #11
    Never mind, I didn't see that you had already tried that.
  13. Nov 22, 2014 #12


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    Staff Emeritus
    Science Advisor
    Gold Member

    Click "info", "help/how-to" and "latex primer"...or use this link: https://www.physicsforums.com/help/latexhelp/
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