<p>Let <em>p</em> = \(\lim_{x \to 0^+} (1 + \tan^2 \sqrt{x})^{1/2x}\), then \(\log p\) is equal to</p>
Step-by-Step Solution
Key Concept: Rewrite the expression using the logarithm property: take log of the limit, then use the standard form lim[u→0] (1+u)^(1/u) = e with careful substitution of u = tan²√x.
<p><strong>Step 1:</strong> Take logarithm to convert the exponential form:</p><p>log p = lim_{x→0⁺} (1/2x) · log(1 + tan²√x)</p><p><strong>Step 2:</strong> Rewrite using limit algebra:</p><p>log p = (1/2) · lim_{x→0⁺} [log(1 + tan²√x)]/x</p><p><strong>Step 3:</strong> Use the substitution u = √x, so x = u² and as x→0⁺, u→0⁺:</p><p>log p = (1/2) · lim_{u→0⁺} [log(1 + tan²u)]/u²</p><p><strong>Step 4:</strong> Use the standard limit log(1 + t)≈t as t→0:</p><p>log p = (1/2) · lim_{u→0⁺} tan²u/u²</p><p><strong>Step 5:</strong> Apply lim_{u→0} (tan u)/u = 1:</p><p>log p = (1/2) · [lim_{u→0⁺} (tan u/u)]² = (1/2) · 1² = 1/2</p><p><strong>∴ Answer: log p = 1/2</strong></p>
Correct Answer: D