Question

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

Answer 1

$\underset{x\to 0}{\lim }\left[\frac{1-\cos \left(mx\right)}{x^2}\right]$

$$\begin{gathered} =\underset{x\to 0}{\lim }\left[\frac{2{\sin }^2\left({\displaystyle \frac{mx}{2}}\right)}{x^2}\right]\left[\because 1-\cos A=2{\sin }^2\left(\frac{A}{2}\right)\right] \\ =2\underset{x\to 0}{\lim }\left[\frac{\sin \left({\displaystyle \frac{mx}{2}}\right)}{{\displaystyle \frac{mx}{2}}}\times \frac{\sin \left({\displaystyle \frac{mx}{2}}\right)}{{\displaystyle \frac{mx}{2}}}\right]\times \frac{m}{2}\times \frac{m}{2} \\ =2\times \frac{m}{2}\times \frac{m}{2}\left[\because \underset{x\to 0}{\lim }\frac{\sin x}{x}=1\right] \\ =\frac{m^2}{2} \end{gathered}$$