Question

The value of the limit $\underset{x\to \frac{\pi }{2}}{\lim }\frac{4\sqrt{2}\left(\sin \left(3x\right)+\sin \left(x\right)\right)}{\left(2\sin \left(2x\right)\sin \left({\displaystyle \frac{3x}{2}}\right)+\cos \left({\displaystyle \frac{5x}{2}}\right)\right)-\left(\sqrt{2}+\sqrt{2}\cos \left(2x\right)+\cos \left({\displaystyle \frac{3x}{2}}\right)\right)}$ is _______________

Answer 1

Find the value of the given limit

$$\begin{gathered} \underset{x\to \frac{\pi }{2}}{\lim }\frac{4\sqrt{2}\left(\sin \left(3x\right)+\sin \left(x\right)\right)}{\left(2\sin \left(2x\right)\sin \left({\displaystyle \frac{3x}{2}}\right)+\cos \left({\displaystyle \frac{5x}{2}}\right)\right)-\left(\sqrt{2}+\sqrt{2}\cos \left(2x\right)+\cos \left({\displaystyle \frac{3x}{2}}\right)\right)} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{4\sqrt{2}\left(\sin \left(3x\right)+\sin \left(x\right)\right)}{\left(\cos \left({\displaystyle \frac{x}{2}}\right)-\cos \left({\displaystyle \frac{7x}{2}}\right)+\cos \left({\displaystyle \frac{5x}{2}}\right)\right)-\left(\sqrt{2}+\sqrt{2}\left(2{\cos }^2\left(x\right)-1\right)+\cos \left({\displaystyle \frac{3x}{2}}\right)\right)}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{)}\mathbf{-}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{)}\mathbf{,}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{2}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{2}}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{-}\mathbf{1}\mathbf{]} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{4\sqrt{2}\left(\sin \left(3x\right)+\sin \left(x\right)\right)}{\left(\cos \left({\displaystyle \frac{x}{2}}\right)-\cos \left({\displaystyle \frac{3x}{2}}\right)\right)+\left(\cos \left({\displaystyle \frac{5x}{2}}\right)-\cos \left({\displaystyle \frac{7x}{2}}\right)\right)-2\sqrt{2}{\cos }^2\left(x\right)} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{4\sqrt{2}\left(2\sin \left(2x\right)\cos \left(x\right)\right)}{\left(2\sin \left(x\right)\sin \left({\displaystyle \frac{x}{2}}\right)\right)+\left(2\sin \left(3x\right)\sin \left({\displaystyle \frac{x}{2}}\right)\right)-2\sqrt{2}{\cos }^2\left(x\right)}\mathbf{[}\mathbf{\because }\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{-}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\frac{\mathbf{y}\mathbf{+}\mathbf{x}}{\mathbf{2}}\mathbf{\right)}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\frac{\mathbf{y}\mathbf{-}\mathbf{x}}{\mathbf{2}}\mathbf{\right)}\mathbf{,}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\frac{\mathbf{x}\mathbf{+}\mathbf{y}}{\mathbf{2}}\mathbf{\right)}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\frac{\mathbf{x}\mathbf{-}\mathbf{y}}{\mathbf{2}}\mathbf{\right)}\mathbf{]} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{8\sqrt{2}\left(\sin \left(2x\right)\cos \left(x\right)\right)}{2\sin \left({\displaystyle \frac{x}{2}}\right)\left(\sin \left(x\right)+\sin \left(3x\right)\right)-2\sqrt{2}{\cos }^2\left(x\right)} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{8\sqrt{2}\left(\sin \left(2x\right)\cos \left(x\right)\right)}{2\sin \left({\displaystyle \frac{x}{2}}\right)\left(2\sin \left(2x\right)\cos \left(x\right)\right)-2\sqrt{2}{\cos }^2\left(x\right)}\mathbf{[}\mathbf{\because }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{+}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\frac{\mathbf{x}\mathbf{+}\mathbf{y}}{\mathbf{2}}\mathbf{\right)}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\frac{\mathbf{y}\mathbf{-}\mathbf{x}}{\mathbf{2}}\mathbf{\right)}\mathbf{]} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{16\sqrt{2}\left(\sin \left(x\right){\cos }^2\left(x\right)\right)}{4\sin \left({\displaystyle \frac{x}{2}}\right){\cos }^2\left(x\right)\left(2\sin \left(x\right)\right)-2\sqrt{2}{\cos }^2\left(x\right)}\mathbf{[}\mathbf{\because }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{2}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{]} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\frac{16\sqrt{2}\sin \left(x\right)}{8\sin \left({\displaystyle \frac{x}{2}}\right)\sin \left(x\right)-2\sqrt{2}} \\ =\frac{16\sqrt{2}}{8{\displaystyle \frac{1}{\sqrt{2}}}-2\sqrt{2}}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{\left(}\frac{\mathbf{\pi }}{\mathbf{4}}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\mathbf{}\mathbf{]} \\ =\frac{32}{8-4} \\ =8 \end{gathered}$$Hence the value of the limit is $8$