<p>We have
\[\lim_{x \to 1} \left(1 + \frac{a}{x} - \frac{4}{x^2}\right)^{2x} = e^3\]
Find the value of \(a\).</p>
Step-by-Step Solution
Key Concept: Rewrite the expression in the form (1 + u)^v where u → 0, then use the limit formula lim(1+u)^(v/u) = e^(lim v·u). Here, express the base as 1 + (a/x - 4/x²) and use e^(lim 2x·(a/x - 4/x²)) to extract the exponent.
<p><strong>Step 1:</strong> Recognize this as an indeterminate form of type 1^∞. Use the standard formula: if lim f(x) = 1 and lim g(x)·ln(f(x)) = L, then lim[f(x)]^g(x) = e^L.</p><p><strong>Step 2:</strong> Here f(x) = 1 + a/x - 4/x² and g(x) = 2x. We need:</p><p>lim_{x→1} 2x·ln(1 + a/x - 4/x²) = 3</p><p><strong>Step 3:</strong> As x→1, we have a/x - 4/x² → (a - 4). For the limit to exist and equal 3, use ln(1+u) ≈ u for small u:</p><p>lim_{x→1} 2x·(a/x - 4/x²) = 3</p><p><strong>Step 4:</strong> Simplify the exponent term:</p><p>lim_{x→1} [2x·a/x - 2x·4/x²] = lim_{x→1} [2a - 8/x]</p><p><strong>Step 5:</strong> As x→1:</p><p>2a - 8 = 3</p><p>2a = 11</p><p>a = 11/2</p><p><strong>Verification:</strong> At x=1, the exponent becomes 2(1)(11/2 - 4) = 2(3/2) = 3 ✓</p><p>∴ Answer: A (a = 11/2)</p>
Correct Answer: A