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

To evaluate the limit:

\[\lim_{x \to 0} \csc{x} \left( \sqrt{2 \cos^2{x} + 3 \cos{x}} - \sqrt{\cos^2{x} + \sin{x} + 4} \right)\]

We analyze each component as \( x \to 0 \):

1. \( \csc{x} = \frac{1}{\sin{x}} \). As \( x \to 0 \), \( \sin{x} \approx x \), so \( \csc{x} \approx \frac{1}{x} \).
2. Examine terms under the square roots:
   • \( 2\cos^2{x} + 3\cos{x} \to 2(1)^2 + 3(1) = 5 \) as \( x \to 0 \).
   • \( \cos^2{x} + \sin{x} + 4 \to 1 + 0 + 4 = 5 \) as \( x \to 0 \).
3. Using approximations for small \( x \):
   • \( \sqrt{2\cos^2{x} + 3\cos{x}} \approx \sqrt{5} \).
   • \( \sqrt{\cos^2{x} + \sin{x} + 4} \approx \sqrt{5 + x} \approx \sqrt{5}\left(1 + \frac{x}{10}\right) \).
4. The expression becomes:

\[\lim_{x \to 0} \frac{1}{x} \left( \sqrt{5} - \sqrt{5}\left(1 + \frac{x}{10}\right) \right) = \lim_{x \to 0} \frac{1}{x} \left(\sqrt{5} - \sqrt{5} - \frac{\sqrt{5} x}{10}\right)\]

1. Simplifying yields:

\[\lim_{x \to 0} \left(-\frac{\sqrt{5}}{10}\right) = -\frac{\sqrt{5}}{10}\]

Thus, the limit evaluates to:

\[\lim_{x \to 0} = -\frac{1}{2\sqrt{5}}\]

The correct answer is \(- \frac{1}{2\sqrt{5}}\).