Finding value of limit involving cosine I know we can use Maclaurin expansion and l'hopital's rule for solving it but I want another way . Find value of $\lim_{x \to 0} \frac{\cos^2x - \sqrt{\cos x}}{x^2}$.
My try : I multiplied and then divided by conjugate of numerator but it didn't help .

Two standard limits $$\lim_{x\to a} \frac{x^{n} - a^{n}} {x-a} =na^{n-1},\,\lim_{x\to 0}\frac{1-\cos x} {x^{2}}=\frac{1}{2}$$ come to our rescue here. We have
\begin{align} 
L&=\lim_{x\to 0}\frac{\cos^{2}x-\sqrt{\cos x}} {x^{2}}\notag\\
&=\lim_{x\to 0}\frac{\cos x - 1}{x^{2}}\cdot\left(\frac{\cos^{2}x-1}{\cos x - 1}-\frac{\cos^{1/2}x - 1}{\cos x - 1}\right)\notag\\
&=-\frac{1}{2}\left(2-\frac{1}{2}\right)\notag\\
&=-\frac{3}{4}\notag
\end{align} 
