<p>If \(\lim_{x \to c} f(x) \cdot g(x)\) exists then both \(\lim_{x \to c} f(x)\) and \(\lim_{x \to c} g(x)\) exist.</p><p><em>State whether this statement is true or false.</em></p>
Step-by-Step Solution
Key Concept: A product limit can exist even when individual limits don't exist—one function can oscillate while the other approaches zero, making their product converge. The converse of a limit rule is not always true.
<p><strong>Step 1:</strong> Recall the standard limit rule: If lim[x→c] f(x) = L and lim[x→c] g(x) = M both exist, then lim[x→c] f(x)·g(x) = L·M exists.</p><p><strong>Step 2:</strong> Test the converse with a counterexample:<br>Let f(x) = sin(1/x) and g(x) = x as x→0.<br>• lim[x→0] sin(1/x) does NOT exist (oscillates between -1 and 1)<br>• lim[x→0] x = 0 exists<br>• But lim[x→0] x·sin(1/x) = 0 exists (bounded function × vanishing function)</p><p><strong>Step 3:</strong> Since we found a counterexample where the product limit exists but individual limits don't both exist, the statement is <strong>FALSE</strong>.</p><p>∴ Answer: B (False)</p>
Correct Answer: B