lim/n->oo(√(n)*(√(n^2+1)-n))

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

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

.