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 begin by examining each component of the expression as \( x \to 0 \).

1. \( \csc{x} = \frac{1}{\sin{x}} \). As \( x \to 0 \), \( \sin{x} \approx x \), so \( \csc{x} \) behaves like \( \frac{1}{x} \).
2. Consider the terms inside the square roots:
   • \( 2\cos^2{x} + 3\cos{x} \) when \( x \to 0 \): since \( \cos{x} \approx 1 \), we have \( 2(1)^2 + 3(1) = 5 \).
   • \( \cos^2{x} + \sin{x} + 4 \) when \( x \to 0 \): with \( \cos{x} \approx 1 \) and \( \sin{x} \approx x \), we have \( 1 + x + 4 = 5 + x \).
3. Using first-order approximations for small \( x \),
   • \( \sqrt{2\cos^2{x} + 3\cos{x}} \approx \sqrt{5} \) since \( \cos{x} \approx 1 \).
   • \( \sqrt{\cos^2{x} + \sin{x} + 4} \approx \sqrt{5 + x} \approx \sqrt{5}\left(1 + \frac{x}{10}\right) \).
4. The expression becomes: 

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

1. Simplifying, we have: 

\[-\frac{\sqrt{5}}{10} \quad \text{as} \quad x \to 0\]

Thus, the limit evaluates to:

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

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

Answer 2

We are asked to evaluate the limit: 

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

Step 1: Simplifying the Expression

The expression involves two square roots. To simplify the difference of square roots, we use the identity:

\( \sqrt{A} - \sqrt{B} = \frac{A - B}{\sqrt{A} + \sqrt{B}}. \)

Let \( A = 2 \cos^2{x} + 3 \cos{x} \) and \( B = \cos^2{x} + \sin{x} + 4 \). Thus, we can rewrite the original limit as:

\( \lim_{x \to 0} \csc{x} \left( \frac{A - B}{\sqrt{A} + \sqrt{B}} \right) \)

Step 2: Expanding \( A - B \)

Now, calculate \( A - B \):

\( A - B = (2 \cos^2{x} + 3 \cos{x}) - (\cos^2{x} + \sin{x} + 4) \)

Simplifying this expression: \[ A - B = 2 \cos^2{x} + 3 \cos{x} - \cos^2{x} - \sin{x} - 4 \] \[ A - B = \cos^2{x} + 3 \cos{x} - \sin{x} - 4 \]

Step 3: Expanding and Substituting at \( x \to 0 \)

Now, as \( x \to 0 \), we use the small angle approximations:

• \( \cos{x} \approx 1 \)
• \( \sin{x} \approx x \)

Substituting these approximations in \( A - B \): \[ A - B \approx 1 + 3(1) - x - 4 = 0 - x. \]

Step 4: Simplifying the Square Root Expression

Now, we approximate the denominator \( \sqrt{A} + \sqrt{B} \) at \( x \to 0 \). Using the small angle approximation again: \[ \sqrt{2 \cos^2{x} + 3 \cos{x}} \approx \sqrt{2 + 3} = \sqrt{5}, \] \[ \sqrt{\cos^2{x} + \sin{x} + 4} \approx \sqrt{1 + 0 + 4} = \sqrt{5}. \] Therefore, \( \sqrt{A} + \sqrt{B} \approx 2\sqrt{5} \).

Step 5: Final Expression

Now, substitute all the approximations into the original expression: \[ \lim_{x \to 0} \csc{x} \left( \frac{0 - x}{2 \sqrt{5}} \right). \] Since \( \csc{x} = \frac{1}{\sin{x}} \approx \frac{1}{x} \) as \( x \to 0 \), we get: \[ \lim_{x \to 0} \frac{1}{x} \times \frac{-x}{2 \sqrt{5}} = -\frac{1}{2\sqrt{5}}. \]

Answer:

The value of the limit is: \( - \frac{1}{2\sqrt{5}} \).