lim/n->oo(√(n)*(√(n^2+1)-n))

$$\small{\text{ $ \lim \limits_{n \rightarrow \infty } \sqrt{n}\cdot \left( \sqrt{n^2+1}-n \right) $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \sqrt{n}\cdot \left( \sqrt{\frac{n^2}{n^2}\cdot(n^2+1)}-n \right) $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \sqrt{n}\cdot \left(n\cdot \sqrt{ 1+\frac{1}{n^2} }-n \right) $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \sqrt{n}\cdot n\cdot \left( \sqrt{ 1+\frac{1}{n^2} }-1 \right) $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \sqrt{n}\cdot n\cdot \left( \sqrt{ 1+\frac{1}{n^2} } - 1 \right) \dfrac{ \sqrt{ 1+\frac{1}{n^2} } + 1 } { \sqrt{ 1+\frac{1}{n^2} } + 1 } $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \sqrt{n}\cdot n\cdot \left( 1+\frac{1}{n^2} - 1^2 \right) \dfrac{ 1 } { \sqrt{ 1+\frac{1}{n^2} } + 1 } $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \dfrac{ \sqrt{n}\cdot n } { n^2 } \cdot \dfrac{ 1 } { \sqrt{ 1+\frac{1}{n^2} } + 1 } $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \dfrac{ \dfrac{ \sqrt{n} } { n } } { \sqrt{ 1+\frac{1}{n^2} } + 1 } $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \dfrac{ \sqrt{ \dfrac{ n } { n^2 } } } { \sqrt{ 1+\frac{1}{n^2} } + 1 } $}}$$

$$\\= \small{\text{ $ \lim \limits_{n \rightarrow \infty } \dfrac{ \sqrt{ \dfrac{ 1} { n } } } { \sqrt{ 1+\frac{1}{n^2} } + 1 } $}}\\\\ = \small{\text{ $ \lim \limits_{n \rightarrow \infty } \dfrac{ \sqrt{ \dfrac{ 1} { n } } } { \sqrt{ 1+ (\frac{1}{n})^2 } + 1 } $}} \qquad | \qquad \boxed{~ \lim \limits_{n \rightarrow \infty } \dfrac{1}{n}=0~}\\\\ = \small{\text{ $ \dfrac{ \sqrt{0} } { \sqrt{ 1 + 0^2 } + 1 } $}}\\\\ = \small{\text{ $ \dfrac{ 0 } { \sqrt{ 1 } + 1 } $}}\\\\ = \small{\text{ $ \dfrac{ 0 } { 1 + 1 } $}}\\\\ = \small{\text{ $ \dfrac{ 0 } { 2 } $}}\\\\ = \small{\text{$ 0 $}}$$