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Math Help - Find the limit

  1. #1
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    Find the limit

    Show that lim x--> infinity square root (x^2+x) -x = 1/2

    note the x^2 + x is under the root only

    Here's what I did so far:

    [square root (x^2+x) -x] [square root (x^2+x) +x] / [square root (x^2+x) +x]

    If we expand it out it becomes:

    x^2+x + x[square root (x^2+x)] - x[square root (x^2+x)] - x^2 / [square root (x^2+x) +x]


    1/ [square root (x^2+x) +x]

    annnd that's where i lose it.. (if i ever had it from the start)
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  2. #2
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    Quote Originally Posted by alisheraz19 View Post
    Show that lim x--> infinity square root (x^2+x) -x = 1/2

    note the x^2 + x is under the root only

    Here's what I did so far:

    [square root (x^2+x) -x] [square root (x^2+x) +x] / [square root (x^2+x) +x]

    If we expand it out it becomes:

    x^2+x + x[square root (x^2+x)] - x[square root (x^2+x)] - x^2 / [square root (x^2+x) +x]


    1/ [square root (x^2+x) +x]

    annnd that's where i lose it.. (if i ever had it from the start)

    The last expression is wrong: it must have x in the numerator. No wonder, since you expanded the product in the numerator in a terrible way: didn't you pay attention to the fact that you had an expression of the very easy form (a-b)(a+b)=a^2-b^2= difference of squares?
    Fix this, and then multiply the whole expression by 1=\frac{\frac{1}{x}}{\frac{1}{x}} so that you'll be able to divide each term in numerator and denominator by x (and get the x into the square root as x^2...!)

    Tonio
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  3. #3
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    I'm not quite following, the expression I have so far is :

    x / square root (x^2+x) + x note* x^2+x is under the root only

    i don't understand how to get 1/2 from here.. i expanded it out that way because i'm not very good with remembering rules ..

    where do i go from here? how does multiplying by 1 = 1/x / 1/x ?
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by alisheraz19 View Post
    I'm not quite following, the expression I have so far is :

    x / square root (x^2+x) + x note* x^2+x is under the root only

    i don't understand how to get 1/2 from here.. i expanded it out that way because i'm not very good with remembering rules ..

    where do i go from here? how does multiplying by 1 = 1/x / 1/x ?
    \frac{1/x}{1/x}\cdot\frac{x}{\sqrt{x^2+x}+x}=\frac{1}{\frac{1}{  x}(\sqrt{x^2+x}+x)}=\frac{1}{\sqrt{\frac{x^2+x}{x^  2}}+1}

    Do you understand what to do from here?
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