Question

Evaluate the limit: \[ \lim_{\theta \to \frac{\pi}{2}} \frac{8\tan^4\theta + 4\tan^2\theta + 5}{(3 - 2\tan\theta)^4} \]

• A. \( -\frac{1}{2} \)
• B. \( \frac{1}{2} \)
• C. \( -4 \)
• D. \( 1 \)

Hint:

Use small angle approximations for trigonometric limits. - Convert limits involving tangent to reciprocal functions for easier computation.

Answer 1

The Correct Option is B

Step 1: Define substitution
Let \( x = \theta - \frac{\pi}{2} \), then as \( \theta \to \frac{\pi}{2} \), we use the approximation: \[ \tan\theta \approx \frac{1}{x} \quad \text{as } x \to 0. \] Step 2: Substitute and Simplify
Substituting in the given expression: \[ 8\tan^4\theta + 4\tan^2\theta + 5 = 8\left(\frac{1}{x}\right)^4 + 4\left(\frac{1}{x}\right)^2 + 5 \] \[ = \frac{8}{x^4} + \frac{4}{x^2} + 5. \] For the denominator: \[ (3 - 2\tan\theta)^4 = (3 - 2\cdot\frac{1}{x})^4 = \left(\frac{3x - 2}{x}\right)^4. \] Step 3: Compute the limit
Taking the limit, the dominant terms give: \[ \lim_{x \to 0} \frac{\frac{8}{x^4} + \frac{4}{x^2} + 5}{\left(\frac{3x - 2}{x}\right)^4} = \frac{8}{16} = \frac{1}{2}. \] Thus, the correct answer is \( \boxed{\frac{1}{2}} \).