Question

$\underset{\theta \to 0}{\lim }\frac{1-\cos \theta }{2{\theta }^2}$

Answer 1

$$\begin{gathered} \underset{\mathrm{\theta }\to 0}{\lim }\left[\frac{1-\cos \mathrm{\theta }}{2{\mathrm{\theta }}^2}\right] \\ =\underset{\mathrm{\theta }\to 0}{\lim }\left[\frac{2{\sin }^2\left({\displaystyle \frac{\mathrm{\theta }}{2}}\right)}{2{\mathrm{\theta }}^2}\right] \\ =\underset{\mathrm{\theta }\to 0}{\lim }\left[\frac{{\sin }^2\left({\displaystyle \frac{\mathrm{\theta }}{2}}\right)}{{\displaystyle \frac{{\mathrm{\theta }}^2}{4}}\times 4}\right] \\ =\underset{\mathrm{\theta }\to 0}{\lim }\left[{\left\{\frac{\sin \left({\displaystyle \frac{\mathrm{\theta }}{2}}\right)}{{\displaystyle \frac{\mathrm{\theta }}{2}}}\right\}}^2\times \frac{1}{4}\right] \\ =\frac{1}{4} \end{gathered}$$