Evaluate `lim_(xto0) (1-cosmx)/(1-cosnx).`

Explanation

To evaluate the limit

\[ \lim_{x \to 0} \frac{1 - \cos(mx)}{1 - \cos(nx)}, \]

we can use the known limit:

\[ \lim_{x \to 0} \frac{1 - \cos(kx)}{(kx)^2} = \frac{1}{2} \quad \text{for any constant } k. \]

### Step-by-Step Solution:

1. **Rewrite the Limit**: Start by rewriting the limit in a form that allows us to use the known limit formula. We can multiply and divide both the numerator and the denominator by \(mx^2\) and \(nx^2\) respectively:

\[ \lim_{x \to 0} \frac{1 - \cos(mx)}{1 - \cos(nx)} = \lim_{x \to 0} \frac{(1 - \cos(mx)) \cdot (mx^2)}{(1 - \cos(nx)) \cdot (nx^2)} \cdot \frac{mx^2}{nx^2}. \]

2. **Apply the Limit Formula**: Now we can apply the limit formula to both the numerator and the denominator:

\[ \lim_{x \to 0} \frac{1 - \cos(mx)}{(mx)^2} = \frac{1}{2} \quad \text{and} \quad \lim_{x \to 0} \frac{1 - \cos(nx)}{(nx)^2} = \frac{1}{2}. \]

3. **Substituting the Limits**: Substitute these limits into our expression:

\[ \lim_{x \to 0} \frac{1 - \cos(mx)}{1 - \cos(nx)} = \frac{\frac{1}{2} \cdot (mx)^2}{\frac{1}{2} \cdot (nx)^2} = \frac{m^2 x^2}{n^2 x^2}. \]

4. **Canceling Terms**: The \(x^2\) terms cancel out:

\[ \frac{m^2}{n^2}. \]

5. **Final Result**: Therefore, the limit evaluates to:

\[ \lim_{x \to 0} \frac{1 - \cos(mx)}{1 - \cos(nx)} = \frac{m^2}{n^2}. \]

### Final Answer:

\[ \frac{m^2}{n^2}. \]