jee-main 2023 Q71

jee-main · India · session1_25jan_shift1 Sequences and series, recurrence and convergence Convergence proof and limit determination

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

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

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