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  • Evaluate    <img alt="\mathrm{\lim _{n \rightarrow \infty} \int_0^{\frac{\pi}{3}} \frac{\sin ^n x}{\sin ^n x+\cos ^n x} d x}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5C

Evaluate    \mathrm{\lim _{n \rightarrow \infty} \int_0^{\frac{\pi}{3}} \frac{\sin ^n x}{\sin ^n x+\cos ^n x} d x}

Option: 1

\mathrm{\frac{\pi}{12}}


Option: 2

\mathrm{\frac{\pi}{2}}


Option: 3

0


Option: 4

1


Answers (1)

                           \mathrm{\lim _{n \rightarrow \infty} \int_0^{\frac{\pi}{3}} \frac{\sin ^n x}{\sin ^n x+\cos ^n x} d x=\int_0^{\frac{\pi}{3}} \lim _{n \rightarrow \infty} \frac{\sin ^n x}{\sin ^n x+\cos ^n x} d x}

Now depending on the relative size of  \mathrm{\sin x \text { and } \cos x} , we have

                         \mathrm{\lim _{n \rightarrow \infty} \frac{\sin ^n x}{\sin ^n x+\cos ^n x}= \begin{cases}0, & \text { if } 0 \leq x<\frac{\pi}{4} \\ \frac{1}{2}, & \text { if } x=\frac{\pi}{4} \\ 1, & \text { if } \frac{\pi}{4}<x \leq \frac{\pi}{3} .\end{cases}}

So it follows that the limit is  \mathrm{\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} d x=\frac{\pi}{12}} .

 

Posted by

Ramraj Saini

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