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

The Correct Option is B

$\underset{n\to \infty }{\mathrm{Lim}}\frac{0+3+6+9+\dots .n\text{terms}}{\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}}$
$\underset{n\to \infty }{\mathrm{Lim}}\frac{3n(n-1)}{2\left(\sqrt{2n^4+4n+3}-\sqrt{n^4+5n+4}\right)}$
\(=\frac3{2(\sqrt2-1)}=\frac3{2}(\sqrt2+1)\)