Question

\( \lim_{x \to 0} \frac{1 - \cos(mx)}{1 - \cos(nx)} \) is

• A. \( \frac{m^2}{n^2} \)
• B. \( \frac{n^2}{m^2} \)
• C. \( \infty \)
• D. \( -\infty \)
• E. \( 0 \)

Hint:

Use standard small-angle approximations in limits.

Answer 1

The Correct Option is A

Concept: Use approximation: \[ 1 - \cos x \approx \frac{x^2}{2} \quad \text{as } x \to 0 \]

Step 1: Apply approximation to numerator.
\[ 1 - \cos(mx) \approx \frac{(mx)^2}{2} = \frac{m^2 x^2}{2} \]

Step 2: Apply approximation to denominator.
\[ 1 - \cos(nx) \approx \frac{n^2 x^2}{2} \]

Step 3: Substitute into limit.
\[ \frac{\frac{m^2 x^2}{2}}{\frac{n^2 x^2}{2}} \]

Step 4: Cancel common factors.
\[ = \frac{m^2}{n^2} \]

Step 5: Final result.
\[ \frac{m^2}{n^2} \]