Question

$\underset{x\to 0}{\lim }\frac{\sin (\pi {\cos }^{2}x)}{x^2}$ equals to

• A. $-\pi$
• B. $\pi$
• C. $\frac{\pi }{2}$
• D. $-\frac{\pi }{2}$

Answer 1

Answer: (B)

The equation of normal at $\left(ct,\frac{c}{t}\right)$ on the curve $xy=c^2$ is $ty=t^{3}x-c{t}^4+c$ If it passes through $\left(ct,\frac{c}{t}\right)$ then $t\frac{c}{t}=t^3.ct-c{t}^4+c$ $\Rightarrow$ $t=t^{3}t{}^2-t^{4}t+t$ $\Rightarrow$ $t-t=t^{3}t(t-t)$ $\Rightarrow$ $1=-t^{3}t$ $\Rightarrow$ $t=-\frac{1}{t^3}$