The sequence limit is:

$$\lim_{n\to \infty}\left[\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n+1}-\sqrt{n}}\right]$$

I rationalized and got:

$$\lim_{n\to \infty}\left[\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n}-\sqrt{n-1}}\right]$$ After this procedure I got stuck

  • $\begingroup$ I would guess you can either squeze it or do conjugate rule. $\endgroup$ – mathreadler Oct 7 '18 at 15:52
  • 2
    $\begingroup$ But the denomiator still constains square roots. What have you done? $\endgroup$ – gammatester Oct 7 '18 at 15:52

Hint. It should be $$\frac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n+1}-\sqrt{n}}=\frac{\sqrt{n+1}+\sqrt{n}}{\sqrt{n}+\sqrt{n-1}}=\frac{\sqrt{1+\frac{1}{n}}+1}{1+\sqrt{1-\frac{1}{n}}}.$$ Can you take it from here?


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