Question

Evaluate the limit: \[ \lim\limits_{x \to 0} \frac{\cos 2x - \cos 3x}{\cos 4x - \cos 5x}.= \]

• A. \( \frac{5}{9} \)
• B. \( \frac{3}{4} \)
• C. \( \frac{2}{5} \)
• D. \( \frac{4}{5} \)

Hint:

For limits involving cosine differences, use the identity: \[ \cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right), \] and apply the small-angle approximation \( \sin y \approx y \) for small \( y \).

Answer 1

The Correct Option is A

Step 1: Use the Cosine Difference Formula
Using the standard trigonometric identity: \[ \cos A - \cos B = -2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right), \] we apply this to both numerator and denominator:
For the numerator: \[ \cos 2x - \cos 3x = -2 \sin \left(\frac{2x + 3x}{2} \right) \sin \left(\frac{2x - 3x}{2} \right). \] \[ = -2 \sin \left(\frac{5x}{2} \right) \sin \left(\frac{-x}{2} \right). \] For the denominator: \[ \cos 4x - \cos 5x = -2 \sin \left(\frac{4x + 5x}{2} \right) \sin \left(\frac{4x - 5x}{2} \right). \] \[ = -2 \sin \left(\frac{9x}{2} \right) \sin \left(\frac{-x}{2} \right). \] Step 2: Apply the Limit
Canceling common terms: \[ \lim\limits_{x \to 0} \frac{\sin \left(\frac{5x}{2} \right)}{\sin \left(\frac{9x}{2} \right)}. \] Using the small-angle approximation \( \sin y \approx y \) as \( y \to 0 \): \[ \frac{\frac{5x}{2}}{\frac{9x}{2}} = \frac{5}{9}. \] Step 3: Conclusion
Thus, the correct answer is: \[ \mathbf{\frac{5}{9}}. \]