Question

${lim}_{x\to a}\frac{cosx-cosa}{\sqrt{x}-\sqrt{a}}$

Answer 1

${lim}_{x\to a}\frac{cosx-cosa}{\sqrt{x}-\sqrt{a}}$

$={lim}_{x\to a}\frac{(-2sin(\frac{x+a}{2}\left)sin\right(\frac{x-a}{2}\left)\right)\times (\sqrt{x}+\sqrt{a})}{(\sqrt{x}-\sqrt{a})(\sqrt{x}+\sqrt{a})}$

$=-2{lim}_{x\to a}\frac{sin\left(\frac{x+a}{2}\right)sin\left(\frac{x-a}{2}\right)\times (\sqrt{x}+\sqrt{a})}{(x-a)}$

$=-2{lim}_{x\to a}sin\left(\frac{x+a}{2}\right)\times {lim}_{x\to a}\frac{sin\left(\frac{x-a}{2}\right)}{\left(\frac{x-a}{2}\right)}\times \frac{1}{2}$

${lim}_{x\to a}(\sqrt{x}+\sqrt{a})$

$=-2sin\left(a\right)\times 1\times \frac{1}{2}\times 2\sqrt{a}[\because {lim}_{\theta \to 0}\frac{sin\theta }{\theta }=1]$

$=-2\sqrt{a}sina$