Question

$$\begin{gathered} 1.\underset{x\to 0}{\lim }\frac{\sin (\mathrm{\pi }{\cos }^2(\tan (\sin \mathrm{x})\left)\right)}{x^2}isequalto: \\ \left(a\right)\mathrm{\pi }\left(\mathrm{b}\right)\frac{\mathrm{\pi }}{4} \\ \left(\mathrm{c}\right)\frac{\mathrm{\pi }}{2}\left(\mathrm{d}\right)\mathrm{None}\mathrm{of}\mathrm{these} \end{gathered}$$

Answer 1

$$\begin{gathered} \underset{x\to 0}{\lim }\frac{\sin \left(\mathrm{\pi }{\cos }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}{x^2} \\ Weknow\sin \mathrm{\theta }=\sin \left(\mathrm{\pi }-\mathrm{\theta }\right) \\ \underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\cos }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle x^2}}=\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }-\mathrm{\pi }{\cos }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle x^2}} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }\left[1-{\cos }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right]\right)}}{{\displaystyle x^2}} \\ Use1-{\cos }^2\theta =si{n}^2\theta \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }\left[si{n}^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right]\right)}}{{\displaystyle x^2}} \\ =\underset{x\to 0}{\lim }\left\{\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle x^2}}\times \frac{\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}{\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}\right\} \\ =\underset{x\to 0}{\lim }\left\{\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \frac{\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}{x^2}\right\} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \underset{x\to 0}{\lim }\frac{\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}{x^2} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times \underset{x\to 0}{\lim }\frac{{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}{x^2} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times \underset{x\to 0}{\lim }\left\{\frac{{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}{x^2\times {\tan }^2\left(\sin \mathrm{x}\right)}\times {\tan }^2\left(\sin \mathrm{x}\right)\right\} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times \underset{x\to 0}{\lim }\left\{\frac{{\displaystyle {\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}{{\displaystyle {\tan }^2\left(\sin \mathrm{x}\right)}}\times \frac{{\displaystyle {\tan }^2\left(\sin \mathrm{x}\right)}}{x^2}\right\} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times \underset{x\to 0}{\lim }\frac{{\displaystyle {\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}{{\displaystyle {\tan }^2\left(\sin \mathrm{x}\right)}}\times \underset{x\to 0}{\lim }\frac{{\displaystyle {\tan }^2\left(\sin \mathrm{x}\right)}}{x^2} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times \underset{x\to 0}{\lim }{\left(\frac{{\displaystyle \sin \left(\tan \left(\sin \mathrm{x}\right)\right)}}{{\displaystyle \tan \left(\sin \mathrm{x}\right)}}\right)}^2\times \underset{x\to 0}{\lim }\left(\frac{{\displaystyle {\tan }^2\left(\sin \mathrm{x}\right)}}{x^2\times {\sin }^{2}x}\times {\sin }^{2}x\right) \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times {\left(\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\tan \left(\sin \mathrm{x}\right)\right)}}{{\displaystyle \tan \left(\sin \mathrm{x}\right)}}\right)}^2\times \underset{x\to 0}{\lim }\frac{{\displaystyle {\tan }^2\left(\sin \mathrm{x}\right)}}{s{\mathrm{in}}^{2}x}\times \underset{x\to 0}{\lim }\frac{{\sin }^{2}x}{x^2} \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times {\left(\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\tan \left(\sin \mathrm{x}\right)\right)}}{{\displaystyle \tan \left(\sin \mathrm{x}\right)}}\right)}^2\times \underset{x\to 0}{\lim }{\left(\frac{{\displaystyle \tan \left(\sin \mathrm{x}\right)}}{\sin x}\right)}^2\times \underset{x\to 0}{\lim }{\left(\frac{\sin x}{x}\right)}^2 \\ =\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)\right)}}{{\displaystyle \mathrm{\pi }{\sin }^2\left(\tan \left(\sin \mathrm{x}\right)\right)}}\times \mathrm{\pi }\times {\left(\underset{x\to 0}{\lim }\frac{{\displaystyle \sin \left(\tan \left(\sin \mathrm{x}\right)\right)}}{{\displaystyle \tan \left(\sin \mathrm{x}\right)}}\right)}^2\times {\left(\underset{x\to 0}{\lim }\frac{{\displaystyle \tan \left(\sin \mathrm{x}\right)}}{\sin x}\right)}^2\times {\left(\underset{x\to 0}{\lim }\frac{\sin x}{x}\right)}^2 \\ Use\underset{\theta \to 0}{\lim }\frac{\sin \theta }{\theta }=1and\underset{\theta \to 0}{\lim }\frac{tan\theta }{\theta }=1 \\ =1\times \mathrm{\pi }\times 1\times 1\times 1 \\ =\pi \end{gathered}$$