Question

The value of

• A. 0
• B. 1/2
• C. 2
• D. None of these

Answer 1

Answer: (D)

None of these

Answer 2

We know that I_2n = $\int _ { 0 } ^ { \pi / 2 } \sin ^ { 2 n } x$ dx

= × × ….. × $\frac { 1 } { 2 }$ ×$\frac { \pi } { 2 }$,

I_{2n + 1} = x dx = × × … ×$\frac { 2 } { 3 }$ and

Also, I_{2m + 1} = I_{2m – 1}.

For all x Ī (0, p/2), sin^{2m – 1} x > sin^2m x > sin^{2m + 1} x

Integrating from 0 to p/2, we get I_{2m – 1} ³ I_2m ³ I_{2m + 1}

Whence ³ ³ 1 ... (i)

Also =$\frac { 2 m + 1 } { 2 m }$.

Hence $\lim _ { m \rightarrow \infty }$ = $\lim _ { m \rightarrow \infty }$ $\frac { 2 m + 1 } { 2 m }$ = 1.

From (i) and using sandwitch theorem we have

$\lim _ { m \rightarrow \infty }$ = 1.