Question

Evaluate the limit: \[ \lim_{x \to 0} \csc{x} \left( \sqrt{2 \cos^2{x} + 3 \cos{x}} - \sqrt{\cos^2{x} + \sin{x} + 4} \right) \] is equal to:

• A. 0
• B. \( \frac{1}{2\sqrt{5}} \)
• C. \( \frac{1}{\sqrt{15}} \)
• D. \( - \frac{1}{2\sqrt{5}} \)

Hint:

For limits involving trigonometric functions, use small angle approximations like \( \sin{x} \approx x \) and \( \cos{x} \approx 1 - \frac{x^2}{2} \) to simplify the expression.

Answer 1

The Correct Option is D

Step 1: We are given the following limit: \[ \lim_{x \to 0} \csc{x} \left( \sqrt{2 \cos^2{x} + 3 \cos{x}} - \sqrt{\cos^2{x} + \sin{x} + 4} \right) \] First, let’s expand the terms inside the square roots using small-angle approximations around \( x = 0 \): For small \( x \), we have: \[ \cos{x} \approx 1 - \frac{x^2}{2}, \quad \sin{x} \approx x \] Substituting these approximations into the expression: \[ \sqrt{2 \left(1 - \frac{x^2}{2}\right)^2 + 3\left(1 - \frac{x^2}{2}\right)} - \sqrt{\left(1 - \frac{x^2}{2}\right)^2 + x + 4} \] After simplifying, this expression becomes: \[ \sqrt{2 - x^2 + 3 - 3x^2/2} - \sqrt{1 - x^2 + x + 4} \] Now, taking the limit as \( x \to 0 \) results in the final answer: \[ \lim_{x \to 0} \csc{x} \left( \frac{-1}{2\sqrt{5}} \right) \] Thus, the correct answer is \( - \frac{1}{2\sqrt{5}} \).