Soit \((a_n)_{n\geqslant 0}\) la suite définie par
\(\displaystyle a_n=\frac{(n+3)^{1/2}-n^{1/2}}{(n+1)^{-1/2}}\).
Alors:
- \(\displaystyle \lim_{n\to\infty}a_n=0\)
- \(\displaystyle \lim_{n\to\infty}a_n=+\infty\)
- \(\displaystyle \lim_{n\to\infty}a_n=\frac32\)
- \(\displaystyle \lim_{n\to\infty}a_n=3\)
\[\begin{aligned}
a_n
&=\frac{(n+3)^{1/2}-n^{1/2}}{(n+1)^{-1/2}}\\
&=\sqrt{n+1}\left(\sqrt{n+3}-\sqrt{n}\right)\\
&=\sqrt{n+1}\frac{3}{\sqrt{n+1}+\sqrt{n}}\\
&=\frac{3\sqrt{1+\frac1n}}{\sqrt{1+\frac1n}+1}\,.
\end{aligned}\]
Donc \(\lim_{n\to\infty}a_n=\frac{3}{2}\).