Question:
\( \lim_{x \to 0} \left(1^{\csc^2 x} + 2^{\csc^2 x} + \cdots + n^{\csc^2 x}\right)\sin^2 x \) is equal to
\( \lim_{x \to 0} \left(1^{\csc^2 x} + 2^{\csc^2 x} + \cdots + n^{\csc^2 x}\right)\sin^2 x \) is equal to
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When $\infty \cdot 0$ form appears, analyze behavior term-wise.
Updated On: Apr 23, 2026
- $1$
- $\frac{1}{n}$
- $n$
- $0$
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The Correct Option is C
Solution and Explanation
Concept:
Use approximation:
\[
\csc^2 x \approx \frac{1}{x^2}, \quad \sin^2 x \approx x^2
\]
Step 1: Rewrite expression.
\[ \sum_{k=1}^{n} k^{\csc^2 x} \cdot \sin^2 x \]
Step 2: Use approximation.
\[ k^{\csc^2 x} \approx k^{1/x^2} \]
Step 3: Analyze dominant behavior.
As $x \to 0$, $\sin^2 x \to 0$ and $\csc^2 x \to \infty$ Each term tends to 1 effectively.
Step 4: Sum of terms.
\[ = 1 + 1 + \cdots + 1 = n \] Conclusion:
Limit = $n$
Step 1: Rewrite expression.
\[ \sum_{k=1}^{n} k^{\csc^2 x} \cdot \sin^2 x \]
Step 2: Use approximation.
\[ k^{\csc^2 x} \approx k^{1/x^2} \]
Step 3: Analyze dominant behavior.
As $x \to 0$, $\sin^2 x \to 0$ and $\csc^2 x \to \infty$ Each term tends to 1 effectively.
Step 4: Sum of terms.
\[ = 1 + 1 + \cdots + 1 = n \] Conclusion:
Limit = $n$
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