Question

The value of the $\underset{x\to 0}{lim}\frac{\sin \left(\pi {\cos }^2\right(\pi {\cos }^{2}x\left)\right)}{x^2}$ is equal to

• A. $1$
• B. $\pi /2$
• C. $\pi$
• D. $0$

Answer 1

The correct option is D $0$

$L=\underset{x\to 0}{lim}\frac{\sin \left[\pi \right(1-{\sin }^2\left(\pi {\cos }^{2}x\right)\left)\right]}{x^2}$

$=\underset{x\to 0}{lim}\frac{\sin \left[\pi {\sin }^2\right(\pi (1-{\sin }^{2}x)\left)\right]}{x^2}$

$=\underset{x\to 0}{lim}\frac{\sin \left[\pi {\sin }^2\right(\pi {\sin }^{2}x\left)\right]}{x^2}$

$=\underset{x\to 0}{lim}\frac{\sin \left[\pi {\sin }^2\right(\pi {\sin }^{2}x\left)\right]}{\pi {\sin }^2\left(\pi {\sin }^{2}x\right)}\cdot \frac{\pi {\sin }^2\left(\pi {\sin }^{2}x\right)}{x^2}=\underset{x\to 0}{lim}\frac{\pi {\sin }^2\left(\pi {\sin }^{2}x\right)}{(\pi {\sin }^{2}x{)}^2}\times \frac{(\pi {\sin }^{2}x{)}^2}{x^2}=0$