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Math Help - Solving for a square root limit

  1. #1
    s3a
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    Solving for a square root limit

    When I try to solve limits I find I always get the answer wrong when dealing with square roots.

    Here's an example:

    lim x --> - infinity (sqrt(x^2 + x + 1) + x

    what I do:

    lim x --> - infinity (sqrt(x^2 + x + 1) + x (((sqrt(x^2+x+1) - x)/((sqrt(x^2+x+1) - x))

    lim x --> -infinity (x^2 + x + 1 - x^2)/(sqrt(x^2 + x + 1) - x)

    lim x --> -infinity (x + 1)/(sqrt(x^2 + x + 1) - x)

    lim x --> -infinity (x + 1)/(sqrt(x^2( 1 + 1/x + 1x^2) - x)

    lim x --> -infinity (x/x(1 + 1/x)/(sqrt((x^2)/(x^2)( 1 + 1/x + 1x^2) - x/x)

    so I get 1/sqrt(1-1) = Does not exist

    but the answer is -1/2.

    What am I doing wrong?

    Any help would be greatly appreciated!
    Thanks in advance!
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  2. #2
    MHF Contributor alexmahone's Avatar
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  3. #3
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    Quote Originally Posted by s3a View Post
    When I try to solve limits I find I always get the answer wrong when dealing with square roots.

    Here's an example:

    lim x --> - infinity (sqrt(x^2 + x + 1) + x

    what I do:

    lim x --> - infinity (sqrt(x^2 + x + 1) + x (((sqrt(x^2+x+1) - x)/((sqrt(x^2+x+1) - x))

    lim x --> -infinity (x^2 + x + 1 - x^2)/(sqrt(x^2 + x + 1) - x)

    lim x --> -infinity (x + 1)/(sqrt(x^2 + x + 1) - x)

    lim x --> -infinity (x + 1)/(sqrt(x^2( 1 + 1/x + 1x^2) - x)

    lim x --> -infinity (x/x(1 + 1/x)/(sqrt((x^2)/(x^2)( 1 + 1/x + 1x^2) - x/x)
    Here's your error. \sqrt{x^2}= |x|, not x. And since you are taking x going to negative infinity, \sqrt{x^2}= -x. You should have 1/(\sqrt{1}+ 1)

    so I get 1/sqrt(1-1) = Does not exist

    but the answer is -1/2.

    What am I doing wrong?

    Any help would be greatly appreciated!
    Thanks in advance!
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  4. #4
    Senior Member DeMath's Avatar
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    Quote Originally Posted by s3a View Post
    When I try to solve limits I find I always get the answer wrong when dealing with square roots.

    Here's an example:

    lim x --> - infinity (sqrt(x^2 + x + 1) + x

    what I do:

    lim x --> - infinity (sqrt(x^2 + x + 1) + x (((sqrt(x^2+x+1) - x)/((sqrt(x^2+x+1) - x))

    lim x --> -infinity (x^2 + x + 1 - x^2)/(sqrt(x^2 + x + 1) - x)

    lim x --> -infinity (x + 1)/(sqrt(x^2 + x + 1) - x)

    lim x --> -infinity (x + 1)/(sqrt(x^2( 1 + 1/x + 1x^2) - x)

    lim x --> -infinity (x/x(1 + 1/x)/(sqrt((x^2)/(x^2)( 1 + 1/x + 1x^2) - x/x)

    so I get 1/sqrt(1-1) = Does not exist

    but the answer is -1/2.

    What am I doing wrong?

    Any help would be greatly appreciated!
    Thanks in advance!
    \mathop {\lim }\limits_{x \to  - \infty } \left( {\sqrt {{x^2} + x + 1}  + x} \right) = \mathop {\lim }\limits_{x \to  - \infty } \frac{{\left( {\sqrt {{x^2} + x + 1}  + x} \right)\left( {\sqrt {{x^2} + x + 1}  - x} \right)}}<br />
{{\sqrt {{x^2} + x + 1}  - x}} =

    = \mathop {\lim }\limits_{x \to  - \infty } \frac{{x + 1}}{{\sqrt {{x^2} + x + 1}  - x}} = \mathop {\lim }\limits_{x \to  - \infty } \frac{{x + 1}}{{\left| x \right|\sqrt {1 + \frac{1}{x} + \frac{1}{{{x^2}}}}  - x}} =

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

    = \frac{{1 - 0}}{{ - \sqrt {1 - 0 - 0}  - 1}} = \frac{1}{{ - 1 - 1}} =  - \frac{1}{2}.
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