Discussion Forum

$\lim_{x \rightarrow 0}\frac{\sin(\pi\cos^{2}x)}{x^{2}}$   is equal to 

Answer:

Explanation:

  $\lim_{x \rightarrow 0}\frac{\sin(\pi\cos^{2}x)}{x^{2}}$

   = $\lim_{x \rightarrow 0}\frac{\sin\pi(1-\sin^{2}x)}{x^{2}}$

     = $\lim_{x \rightarrow 0}\frac{\sin(\pi-\pi\sin^{2}x)}{x^{2}}$

  =$\lim_{x \rightarrow 0}\frac{\sin(\pi\sin^{2}x)}{x^{2}}$

                   [ $\because$   $\sin(\pi-\theta)=\sin\theta$ ]

               $=\lim_{x \rightarrow0}\frac{\sin\pi\sin ^{2}x}{\pi\sin^{2}x}\times (\pi)\left[\frac{\sin^{2}x}{x^{2}}\right]$

  =      $\pi$                     [$\lim_{x \rightarrow0}\frac{\sin\theta}{\theta}=1$]