lim-n-n-1-n-

Question Number 75469 by aliesam last updated on 11/Dec/19

\underset{n \to \infty}{l i m} (\sqrt{n + 1} - \sqrt{n})

Commented by MJS last updated on 11/Dec/19

0

because n + 1 \sim n for large n

Commented by mathmax by abdo last updated on 12/Dec/19

w e h a v e \sqrt{n + 1} - \sqrt{n} = \sqrt{n (1 + \frac{1}{n})} - \sqrt{n} = \sqrt{n} \sqrt{1 + \frac{1}{n}} - \sqrt{n}

\sim \sqrt{n} (1 + \frac{1}{2 n}) - \sqrt{n} = \frac{1}{2 \sqrt{n}} w e h a v e {l i m}_{n \to + \infty} \frac{1}{2 \sqrt{n}} = 0 \Rightarrow

{l i m}_{n \to + \infty} \sqrt{n + 1} - \sqrt{n} = 0