Evaluate the following limits:

$=\lim _{x \rightarrow 0} \frac{\sin m x \times \sin m x}{\sin n x \times \sin n x}$

$=\lim _{x \rightarrow 0} \frac{\frac{\sin m x \times \sin m x}{m x \times m x}}{\frac{\sin n x \times \sin n x}{n x \times n x}} \times \frac{m^{2}}{n^{2}}$

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

$\therefore \lim _{x \rightarrow 0} \frac{\sin m x \times \sin m x}{\sin n x \times \sin n x}=\frac{m^{2}}{n^{2}}$