# Thread: find limit of sequence

1. ## 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.

2. Originally Posted by twilightstr
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.
$\displaystyle \lim_{n \rightarrow \infty}(n+ \sqrt{n + 1}) \cdot \frac{n- \sqrt{n + 1}}{n- \sqrt{n + 1}}$

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

can you take this limit?

3. 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

4. Originally Posted by twilightstr
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, $\displaystyle \lim\left(\sqrt{n+\sqrt{n+1}}-\sqrt{n}\right)\frac{\sqrt{n+\sqrt{n+1}}+\sqrt{n}} {\sqrt{n+\sqrt{n+1}}+\sqrt{n}}=\lim\frac{\sqrt{n+1 }}{\sqrt{n+\sqrt{n+1}}+\sqrt{n}}$ $\displaystyle =\lim\frac{1}{\sqrt{\frac{n}{n+1}+\frac{1}{\sqrt{n +1}}}+\sqrt{\frac{n}{n+1}}}=\boxed{\frac{1}{2}}$

Does this make sense?

5. what happened after the second to last step?

6. Originally Posted by twilightstr
what happened after the second to last step?
Divide through, top and bottom, by the square root of n + 1.