[tex]~~\lim \limits_{x \to 0} \left(\dfrac{1- \cos mx }{1- \cos nx}} \right)\\\\\\=\lim \limits_{x \to 0} \left[\dfrac{2\sin^2 \left(\dfrac{mx}{2} \right)}{2 \sin^2 \left(\dfrac{nx}{2}\right)} \right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;\left[1-\cos 2x =2 \sin^2 x\right]\\\\\\[/tex]
[tex]=\lim \limits_{x \to 0} \left[ \dfrac{\sin^2 \left(\dfrac{mx}2 \right)}{\left(\dfrac{mx}2 \right)^2 } } \times \left(\dfrac{mx}{2} \right)^2\times \dfrac{\left(\dfrac{nx}2 \right)^2} {\sin^2 \left(\dfrac{nx}2 \right)} \times \left(\dfrac{2}{nx} \right)^2\right]\\\\\\=\lim \limits_{x \to 0}\left[ \left( \dfrac{mx}2 \right)^2 \left( \dfrac{2}{nx} \right)^2\right]~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~;\left[\lim \limits_{x \to 0} \dfrac{\sin x}{x} = \lim \limits_{x \to 0} \dfrac{x}{ \sin x} = 1\right] \\\\\\[/tex]
[tex]=\lim \limits_{x \to 0} \left( \dfrac{m^2 x^2}{4} \cdot \dfrac{4}{n^2 x^2 } \right)\\\\\\=\lim \limits_{x \to 0} \left( \dfrac{m^2}{n^2}\right)\\ \\\\=\dfrac{m^2}{n^2}[/tex]
