Limits, Continuity & Differentiability
Exponential and Logarithmic Limits
Grade 12
Question:
<p>Let \(p = \lim_{x \to 0^+} (1 + \tan^2 x)^{1/2x}\), then \(\log p\) is equal to</p>
<p>(a) \(2\)</p>
<p>(b) \(1\)</p>
<p>(c) \(\frac{1}{2}\)</p>
<p>(d) \(\frac{1}{4}\)</p>
Step-by-Step Solution
Key Concept: Use logarithmic manipulation and the standard limit $\lim_{x \to 0} \frac{\tan x}{x} = 1$ for exponential limits
<p><strong>Step 1:</strong> Take logarithm: $\log p = \lim_{x \to 0^+} \frac{\log(1 + \tan^2 x)}{2x}$</p><p><strong>Step 2:</strong> Use $\log(1 + u) \approx u$ and $\tan x \approx x$ as $x \to 0$: $\log p = \lim_{x \to 0^+} \frac{\tan^2 x}{2x}$</p><p><strong>Step 3:</strong> Apply $\lim_{x \to 0} \frac{\tan x}{x} = 1$: $\log p = \frac{1}{2}$</p><p>∴ Answer is C.</p>
Correct Answer: C