Evaluate the following limits:

To Find: Limits

NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form.

In this Case, indeterminate Form is $\frac{0}{0}$

Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$

Divide numerator and denominator by $m^{2}$ and $n^{2}$, we have

So, by using the above formula, we have

$\lim _{x \rightarrow 0} \frac{1-\cos m x}{1-\cos n x}=\lim _{x \rightarrow 0} \frac{\frac{m^{2}[1-\cos m x]}{(m x)^{2}}}{\frac{n^{2}[1-\cos n x]}{(n x)^{2}}}=\frac{m^{2}}{n^{2}}$

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