Question

The value of $\displaystyle\lim _{n \rightarrow \infty} \frac{1+2-3+4+5-6+\ldots .+(3 n-2)+(3 n-1)-3 n}{\sqrt{2 n^4+4 n+3-} \sqrt{n^4+5 n+4}}$ is :

• A. $3(\sqrt{2}+1)$
• B. $\frac{3}{2}(\sqrt{2}+1)$
• C. $\frac{\sqrt{2}+1}{2}$
• D. $\frac{3}{2 \sqrt{2}}$

Answer 1

Answer: (B)

$ \underset{n \rightarrow \infty}{Lim} \frac{0+3+6+9+\ldots . n \text { terms }}{\sqrt{2 n^4+4 n+3}-\sqrt{n^4+5 n+4}} $
$ \underset{n \rightarrow \infty}{Lim} \frac{3 n(n-1)}{2\left(\sqrt{2 n^4+4 n+3}-\sqrt{n^4+5 n+4}\right)} $
$=\frac{3}{2(\sqrt{2}-1)}=\frac{3}{2}(\sqrt{2}+1)$