Evaluate the following limits:

$=\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}$

$=\lim _{x \rightarrow 0} \frac{\frac{1-\cos m x}{m x \times m x}}{\frac{1-\cos n x}{n x \times n x}} \times \frac{m \times m}{n \times n}$

$=\frac{\mathrm{m}^{2}}{\mathrm{n}^{2}}$

$\therefore \lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}=\frac{m^{2}}{n^{2}}$