Evaluate the following limits:


Evaluate the following limits:

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



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

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


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


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