JEE Mains · Maths · STD 11 - 12. limits
\(\lim _{t \rightarrow 0}\left(1^{\frac{1}{\sin ^2 t}}+2^{\frac{1}{\sin ^2 t}}+3^{\frac{1}{\sin ^2 t}} \ldots \ldots n^{\frac{1}{\sin ^2 t}}\right)^{\sin ^2 t}\) is equal to
- A \(n^2+n\)
- B \(n\)
- C \(\frac{n(n+1)}{2}\)
- D \(n^2\)
Answer & Solution
Correct Answer
(B) \(n\)
Step-by-step Solution
Detailed explanation
- As \(t \rightarrow 0\), we have \(\sin ^2 t \rightarrow 0^{+}\), hence \(\frac{1}{\sin ^2 t} \rightarrow \infty\). - Among the terms \[ 1 \frac{1}{\sin ^2 t}, 2^{\frac{1}{\sin ^2 t}}, \ldots, n^{\frac{1}{\sin ^2 t}}, \] the largest base \(n\) dominates the sum. - So the sum…
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