Question:

<div class="solution"><p><strong>Key Idea:</strong> Use the standard result: if <span class="math-inline">$\lim_{x \to 0} f(x) = 0$</span>, then <span class="math-inline">$\lim_{x \to 0}(1 + f(x))^{g(x)} = e^{\lim f(x) \cdot g(x)}$</span>.</p><p><strong>Step 1:</strong> Identify <span class="math-inline">$f(x) = ax + bx^2$</span> and <span class="math-inline">$g(x) = \dfrac{2}{x}$</span>.</p><p>As <span class="math-inline">$x \to 0$</span>, <span class="math-inline">$f(x) \to 0$</span> ✓</p><p><strong>Step 2:</strong> Compute the exponent:<br><span class="math-block">$$\lim_{x \to 0} f(x) \cdot g(x) = \lim_{x \to 0} (ax + bx^2) \cdot \frac{2}{x} = \lim_{x \to 0} (2a + 2bx) = 2a$$</span></p><p><strong>Step 3:</strong> Set equal to 3:<br><span class="math-block">$$2a = 3 \implies a = \frac{3}{2}$$</span></p><p><strong>Step 4:</strong> The term <span class="math-inline">$bx^2 \cdot \dfrac{2}{x} = 2bx \to 0$</span> regardless of <span class="math-inline">$b$</span>. So <span class="math-inline">$b$</span> can be any real number.</p><p><strong>Answer: (A)</strong> <span class="math-inline">$a = \dfrac{3}{2},\ b \in \mathbb{R}$</span></p><div class="trap-box"><strong>Trap:</strong> Option (B) says <span class="math-inline">$b \in \mathbb{R}^+$</span>. Students think <span class="math-inline">$b$</span> must be positive to keep the base positive near 0. But <span class="math-inline">$b$</span> vanishes completely — it has zero effect on the limit.</div><div class="key-concept"><strong>Key Concept:</strong> <span class="math-inline">$1^\infty$</span> indeterminate form — standard exponential limit</div></div>