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Math Help - find limit of sequence

  1. #1
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    find limit of sequence

    An= square root (n+ square root n + 1) - n find limit of sequence.

    my work: lim as x goes to infinite x(1+ (x+1)^ 1/2 / (x))^ (1/2) - squareroot x

    i think the limit is infinite minus infinite but how do i change it into the indeterminate form.
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  2. #2
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    Quote Originally Posted by twilightstr View Post
    An= square root (n+ square root n + 1) - n find limit of sequence.

    my work: lim as x goes to infinite x(1+ (x+1)^ 1/2 / (x))^ (1/2) - squareroot x

    i think the limit is infinite minus infinite but how do i change it into the indeterminate form.
     \lim_{n \rightarrow \infty}(n+ \sqrt{n + 1}) \cdot \frac{n- \sqrt{n + 1}}{n- \sqrt{n + 1}}

    \lim_{n \rightarrow \infty}\frac{n^2- (n + 1)}{n- \sqrt{n + 1}}

    can you take this limit?
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  3. #3
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    the problem is square root of n + squareroot of n+1)) - square root of n
    everything before square root n is within a square root
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by twilightstr View Post
    the problem is square root of n + squareroot of n+1)) - square root of n
    everything before square root n is within a square root
    Thus, \lim\left(\sqrt{n+\sqrt{n+1}}-\sqrt{n}\right)\frac{\sqrt{n+\sqrt{n+1}}+\sqrt{n}}  {\sqrt{n+\sqrt{n+1}}+\sqrt{n}}=\lim\frac{\sqrt{n+1  }}{\sqrt{n+\sqrt{n+1}}+\sqrt{n}} =\lim\frac{1}{\sqrt{\frac{n}{n+1}+\frac{1}{\sqrt{n  +1}}}+\sqrt{\frac{n}{n+1}}}=\boxed{\frac{1}{2}}

    Does this make sense?
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  5. #5
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    what happened after the second to last step?
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  6. #6
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    Quote Originally Posted by twilightstr View Post
    what happened after the second to last step?
    Divide through, top and bottom, by the square root of n + 1.
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