[tex] n\to \infty \Rightarrow \sqrt{n^2+1} \approx \sqrt{n^2} [/tex]
Astfel:
[tex]\lim\limits_{n\to\infty} \left(\sqrt{n^2+1} - n\sqrt{n}\right) = \lim\limits_{n\to\infty} \left(\sqrt{n^2} - n\sqrt{n}\right) =[/tex]
[tex] = \lim\limits_{n\to\infty} \left(|n| - n\sqrt{n}\right) = \lim\limits_{n\to\infty} \left(n - n\sqrt{n}\right) = \lim\limits_{n\to\infty} n\left(1-\sqrt{n}\right) =[/tex]
[tex] = \infty\cdot (-\infty) = -\infty[/tex]
