<p>Let \(m\) be a positive integer. If \(\displaystyle\lim_{x \to 0} |\cos x + \sin 2x + \sin 3x|^{\cot x} = e^m\), then the value of \(m\) is:</p>
Step-by-Step Solution
Key Concept: For limits of the form f(x)^g(x) where f(x)→1 and g(x)→∞, rewrite as e^(g(x)·ln(f(x))) and use Taylor expansions near x=0. The exponent approaches a finite limit only if the numerator and denominator in the L'Hôpital form balance correctly.
<p><strong>Step 1:</strong> Recognize this is a 1^∞ indeterminate form. Rewrite as:</p><p>L = lim_{x→0} e^(cot x · ln|cos x + sin 2x + sin 3x|)</p><p><strong>Step 2:</strong> Expand using Taylor series near x=0:</p><p>cos x = 1 - x²/2 + x⁴/24 + ...</p><p>sin 2x = 2x - 4x³/3 + ...</p><p>sin 3x = 3x - 9x³/2 + ...</p><p><strong>Step 3:</strong> Sum the terms:</p><p>cos x + sin 2x + sin 3x = 1 + (2x + 3x) + (-x²/2) + (-4x³/3 - 9x³/2) + ...</p><p>= 1 + 5x - x²/2 - 35x³/6 + ...</p><p><strong>Step 4:</strong> Since the expression is positive near x=0, we have:</p><p>ln(1 + 5x - x²/2 - 35x³/6 + ...) = (5x - x²/2 - 35x³/6) - (5x - x²/2)²/2 + ...</p><p>= 5x - x²/2 - 35x³/6 - 25x²/2 + ...</p><p>= 5x - 13x²/2 + ...</p><p><strong>Step 5:</strong> Now compute cot x · ln(...) where cot x = 1/x - x/3 - x³/45 + ...</p><p>cot x · ln(...) = (1/x - x/3 - ...)(5x - 13x²/2 + ...)</p><p>= 5 - 13x/2 - 5x/3 + ... = 5 - (43x/6) + ...</p><p><strong>Step 6:</strong> As x→0, cot x · ln(...) → 5</p><p>∴ lim = e^5, so m = <strong>5</strong></p>
Correct Answer: C