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  • Evaluate   <img alt="\mathrm{\lim _{x \rightarrow \pi / 2} \frac{\sqrt{1+\cos (2 x)}}{\sqrt{\pi}-\sqrt{2 x}}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%

Evaluate   \mathrm{\lim _{x \rightarrow \pi / 2} \frac{\sqrt{1+\cos (2 x)}}{\sqrt{\pi}-\sqrt{2 x}}}

Option: 1

\mathrm{ \sqrt{2 \pi}}


Option: 2

\mathrm{\sqrt{3 \pi}}


Option: 3

\mathrm{0}


Option: 4

\mathrm{1}


Answers (1)

best_answer

Dividing numerator and denominator by \mathrm{\sqrt{2 }}

                                                           \mathrm{\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{\frac{1+\cos 2 x}{2}}}{\sqrt{\frac{\pi}{2}}-\sqrt{x}}}

     Multiplying and dividing by \mathrm{\left(\sqrt{\frac{\pi}{2}}+\sqrt{x}\right)}

                                                                 \mathrm{=\lim _{x \rightarrow \frac{\pi}{2}}\left(\sqrt{\frac{\pi}{2}}+\sqrt{x}\right)\left(\frac{\sqrt{\frac{1+\cos 2 x}{2}}}{\frac{\pi}{2}-x}\right)}

As  \mathrm{\sqrt{\frac{1+\cos 2 x}{2}}=\cos x}

                                \mathrm{\begin{gathered} =\left(\lim _{x \rightarrow \frac{\pi}{2}} \sqrt{\frac{\pi}{2}}+\sqrt{x}\right)\left(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\frac{\pi}{2}-x}\right) \\ =\sqrt{2 \pi} \lim _{x \rightarrow \frac{\pi}{2}} \frac{\sin \left(\frac{\pi}{2}-x\right)}{\frac{\pi}{2}-x} \end{gathered}}

         \mathrm{Let \: t=\frac{\pi}{2}-x}

        \mathrm{x \rightarrow \frac{\pi}{2} \: then \: t \rightarrow 0}

                    \mathrm{\begin{aligned} &=\sqrt{2 \pi} \lim _{t \rightarrow 0} \frac{\sin t}{t}\\ &\lim _{x \rightarrow \frac{\pi}{2}} \frac{\sqrt{1+\cos 2 x}}{\sqrt{\pi}-\sqrt{2 x}}=\sqrt{2 \pi} \end{aligned}}

 

Posted by

sudhir.kumar

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