$\lim _{t \rightarrow 0}\left(1^{\frac{1}{\sin ^{2} t}}+2^{\frac{1}{\sin ^{2} t}}+\ldots .+n^{\frac{1}{\sin ^{2} t}}\right)^{\sin ^{2} t}$ is equal to

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Limits

lim_{t \to 0} (1^{\frac{1}{s i n ^{2} t}} + 2^{\frac{1}{s i n ^{2} t}} + \dots . + n^{\frac{1}{s i n ^{2} t}})^{s i n^{2} t}

is equal to

Answer: (B)

Put

s = \frac{1}{sin ^{2} t}

, which tends to

\infty

t \to 0

; the expression is

(1^{s} + 2^{s} + \dots + n^{s})^{1/ s}

. Factoring out the largest base,

n ((\frac{1}{n})^{s} + \dots + (\frac{n - 1}{n})^{s} + 1)^{1/ s}

, every bracketed term except the last tends to

0

, so the bracket raised to

1/ s

tends to

1

. The limit is

n .

JEE Main · 2023 · 24 Jan 2023 · Shift 2 · Single correct option

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