<p>If \(\lim f(x)\) and \(\lim g(x)\) exist then \(\lim [f(x) \cdot g(x)]\) exist.</p>
Step-by-Step Solution
Key Concept: The product of two limits exists if and only if both individual limits exist independently. This is a fundamental limit theorem: if lim f(x) = L and lim g(x) = M both exist, then lim[f(x)·g(x)] = L·M exists.
<p><strong>Step 1:</strong> Recall the fundamental limit theorem: If lim f(x) = L and lim g(x) = M exist (both finite), then their product limit exists.</p><p><strong>Step 2:</strong> Given that both lim f(x) and lim g(x) exist, by the product rule for limits:</p><p>lim[f(x)·g(x)] = [lim f(x)]·[lim g(x)] = L·M</p><p><strong>Step 3:</strong> Since L and M are finite real numbers, their product L·M is also a finite real number, so the limit exists.</p><p><strong>Step 4:</strong> This is a direct application of the algebraic properties of limits—the product of two existing limits always exists.</p><p>∴ <strong>Answer: A (TRUE)</strong> — The statement is correct.</p>
Correct Answer: A