<p>If \(\lim_{x \to 0} f(x)\) and \(\lim_{x \to 0} g(x)\) exist then \(\lim_{x \to 0} g(x)\) exist. \(\lim_{x \to 0} g(x)\) exist.</p>
Step-by-Step Solution
Key Concept: This question tests understanding of limit algebra: if both lim f(x) and lim g(x) exist as x→0, then their quotient lim[f(x)/g(x)] exists only when lim g(x) ≠ 0. The statement as written appears incomplete or has a printing error, but the core insight is recognizing conditions for existence of limits of combined functions.
<p><strong>Step 1:</strong> Recognize the question structure. Given that lim[x→0] f(x) and lim[x→0] g(x) both exist.</p><p><strong>Step 2:</strong> Recall the quotient rule for limits: If lim[x→0] f(x) = L and lim[x→0] g(x) = M both exist, then lim[x→0] [f(x)/g(x)] = L/M, provided M ≠ 0.</p><p><strong>Step 3:</strong> The existence of individual limits is necessary but NOT sufficient for the quotient limit. We must additionally require that lim[x→0] g(x) ≠ 0.</p><p><strong>Step 4:</strong> The correct statement should be: If lim[x→0] f(x) and lim[x→0] g(x) exist AND lim[x→0] g(x) ≠ 0, then lim[x→0] [f(x)/g(x)] exists.</p><p>∴ Answer: A (The quotient limit exists when the denominator limit is non-zero)</p>
Correct Answer: A