Question

Solve

${lim}_{x\to 0}(\sqrt{2}-\sqrt{\frac{1+\cos \left(x\right)}{x^2}})$

Answer 1

${lim}_{x\to 0}(\sqrt{2}-\sqrt{\frac{1+\cos \left(x\right)}{x^2}})$

$={lim}_{x\to 0}\left(\sqrt{2}\right)-{lim}_{x\to 0}\left(\sqrt{\frac{1+\cos \left(x\right)}{x^2}}\right)$

$=\sqrt{2}-\left(\sqrt{{lim}_{x\to 0}(1+\cos \left(x\right))\cdot lim{}_{x\to 0}\left(\frac{1}{x^2}\right)}\right)$

$=\sqrt{2}-\left(\sqrt{2\times \infty }\right)$

$=-\infty$