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.
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.
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?