<p>If \(\lim_{x \to s} f(x)\) and \(\lim_{x \to s} g(x)\) exist then \(\lim_{x \to s} \frac{f(x)}{g(x)}\) exists.</p>
Step-by-Step Solution
Key Concept: The limit of a quotient equals the quotient of limits only when the denominator's limit is non-zero; if lim g(x) = 0, the quotient limit may not exist even when both individual limits exist.
<p><strong>Step 1:</strong> Recall the quotient rule for limits: If lim_{x→s} f(x) = L and lim_{x→s} g(x) = M, then lim_{x→s} f(x)/g(x) = L/M, provided M ≠ 0.</p><p><strong>Step 2:</strong> The given statement claims the limit exists whenever both f(x) and g(x) have limits. This is <strong>incomplete</strong>.</p><p><strong>Step 3:</strong> <strong>Counterexample:</strong> Let f(x) = x and g(x) = x² as x→0. Both limits exist: lim f(x) = 0 and lim g(x) = 0. But lim f(x)/g(x) = lim(x/x²) = lim(1/x) does not exist as x→0.</p><p><strong>Step 4:</strong> The statement is <strong>FALSE</strong> because it lacks the essential condition that lim_{x→s} g(x) ≠ 0.</p><p>∴ Answer: B (Statement is false/incorrect)</p>
Correct Answer: B