<p>If \(\lim_{x \to s} f(x)\) and \(\lim_{x \to s} g(x)\) exist then \(\lim_{x \to s} g(x)\) exist.</p>
Step-by-Step Solution
Key Concept: This statement is logically incomplete/malformed—it claims if two limits exist, then one of them exists, which is trivially true but meaningless. The intended concept likely tests understanding that limit existence of f and g does NOT guarantee existence of composite limits like lim[f(g(x))] without additional continuity conditions.
<p><strong>Step 1:</strong> Recognize the statement structure. As written, 'If lim f(x) and lim g(x) exist, then lim g(x) exists' is a tautology (trivially true but provides no mathematical content).</p><p><strong>Step 2:</strong> The intended concept tests: Given lim f(x) and lim g(x) exist as x→s, does lim[f(g(x))] exist?</p><p><strong>Step 3:</strong> The answer is NO in general. Counterexample: Let f(x) = 1/x and g(x) = 0 for x ≠ 0. Then lim(x→0) g(x) = 0 and lim(x→0) f(x) DNE, but even if lim f existed, lim[f(g(x))] = lim[f(0)] = lim[1/0] = undefined unless f is continuous at g(s).</p><p><strong>Step 4:</strong> Correct principle: lim(x→s) f(g(x)) exists if lim(x→s) g(x) = L AND f is continuous at L.</p><p>∴ Answer: A (The statement as literally written is trivially true; the deeper truth is that composition requires continuity)</p>
Correct Answer: A