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

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Limits

The value of

Lim_{n \to \infty} \frac{1 + 2 - 3 + 4 + 5 - 6 + \dots + ( 3 n - 2 ) + ( 3 n - 1 ) - 3 n}{2 n ^{4} + 4 n + 3 - n ^{4} + 5 n + 4}

is :

Answer: (C)

Each block of three terms in the numerator gives

(3 k - 2) + (3 k - 1) - 3 k = 3 (k - 1)

, so the numerator equals

\sum_{k = 1}^{n} 3 (k - 1) = \frac{3 n ( n - 1 )}{2}

, which grows like

\frac{3}{2} n^{2}

. In the denominator the leading behaviour is

2 n^{4} - n^{4} = n^{2} (2 - 1)

. Hence the ratio approaches

\frac{3/2}{2 - 1} = \frac{3}{2} (2 + 1)

JEE Main · 2023 · 25 Jan 2023 · Shift 1 · Single correct option

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