Question

Assertion :$\underset{x\to 0}{lim}{\{1^{1/{\sin }^{2}x}+2^{1/{\sin }^{2}x}+3^{1/{\sin }^{2}x}+........+n^{1/{\sin }^{2}x}\}}^{{\sin }^{2}x}=n$ Reason: For $0<a<1,\underset{x\to \infty }{lim}{a}^x=0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect and Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Assertion : Let $\frac{1}{si{n}^{2}x}=y$
$\underset{x\to 0}{lim}[1^{1/si{n}^{2}x}+2^{1/si{n}^{2}x}+....+n^{1/si{n}^{2}x}{]}^{si{n}^{2}x}=\underset{y\to \infty }{lim}[1^y+2^y+....+n^y{]}^{1/y}=\underset{y\to \infty }{lim}n{({\left(\frac{1}{n}\right)}^y+{\left(\frac{2}{n}\right)}^y+....+{\left(\frac{n}{n}\right)}^y)}^{1/y}=n{(0+0+...+1)}^{1/y}=n$
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.