Limits And Derivatives

Q: The value of the limit \$\underset{x \rightarrow \dfrac{\pi}{2}}{\lim} \dfrac{4 \sqrt{2} (\sin 3x + \sin x)}{\left(2 \sin 2x \sin \dfrac{3x}{2} + \cos \dfrac{5x}{2} \right) - \left(\sqrt{2} + \sqrt{2} \cos 2x + \cos \dfrac{3x}{2}\right)}\$ is ______.

\$\underset{x \rightarrow \frac{\pi}{2}}{\lim} \dfrac{4 \sqrt{2} (\sin 3x + \sin x)}{\left(2 \sin 2x \sin \dfrac{3x}{2} + \cos \dfrac{5x}{2} \right) - \left(\sqrt{2} + \sqrt{2} \cos x + \cos \dfrac{3x}{2} \right)}\$\$=\underset{x \rightarrow \frac{\pi}{2}}{\lim} \dfrac{8 \sqrt{2} \sin 2x . \cos x}{\cos \dfrac{x}{2} - \cos \dfrac{7x}{2} + \cos \dfrac{5x}{2} - \left(2 \sqrt{2} \cos^2 x + \cos \dfrac{3x}{2}\right)}\$\$=\underset{x \rightarrow \frac{\pi}{2}}{\lim} \dfrac{4 \sqrt{2}. 2. \sin x \cos x. \cos x}{2 \sin x \sin \dfrac{x}{2} + 2 \sin 3x. \sin \dfrac{x}{2} - 2 \sqrt{2} \cos^2 x}\$\$=\underset{x \rightarrow \frac{\pi}{2}}{\lim} \dfrac{16 \sqrt{2} \sin x. \cos^2 x}{2 \sin \dfrac{x}{2} (2 \sin 2x. \cos x ) - 2 \sqrt{2} \cos^2 x}\$\$=\underset{x \rightarrow \frac{\pi}{2}}{\lim} \dfrac{16 \sqrt{2} \sin x}{2 \sin \dfrac{x}{2} . 4 \sin x - 2 \sqrt{2}} = \dfrac{16 \sqrt{2}}{4 \sqrt{2} - 2 \sqrt{2}} = 8\$.

Question

The value of the limit $x \to \frac{π}{2} lim \frac{4 2 ( sin 3 x + sin x )}{( 2 sin 2 x sin \frac{3 x}{2} + cos \frac{5 x}{2} ) - ( 2 + 2 cos 2 x + cos \frac{3 x}{2} )}$ is ______.

Limits And Derivatives

Maths 3688 Views

The value of the limit x→2π​lim​(2sin2xsin23x​+cos25x​)−(2​+2​cos2x+cos23x​)42​(sin3x+sinx)​ is ______.

Maths
3688 Views

The value of the limit $x \to \frac{π}{2} lim \frac{4 2 ( sin 3 x + sin x )}{( 2 sin 2 x sin \frac{3 x}{2} + cos \frac{5 x}{2} ) - ( 2 + 2 cos 2 x + cos \frac{3 x}{2} )}$ is ______.