<p>If \(x^n = x^2 + x + 1\), then \(\lim_{n \to \infty} x^n\) (as \(n \to \infty\), \(x \to 1\)) equals:</p>
Step-by-Step Solution
Key Concept: As n→∞, the equation x^n = x^2 + x + 1 forces x to approach a fixed point where x^n converges. Since x must satisfy this relation for all large n, we need x^∞ = x^2 + x + 1, which means x must equal the limit value that satisfies this equation.
<p><strong>Step 1:</strong> Recognize that x satisfies x^n = x^2 + x + 1 for each n, and we're told that x → 1 as n → ∞.</p><p><strong>Step 2:</strong> Taking the limit as n → ∞ on both sides of x^n = x^2 + x + 1, with x → 1:</p><p>lim(x^n) = lim(x^2 + x + 1)</p><p><strong>Step 3:</strong> Since x → 1 as n → ∞, we have:</p><p>lim(x^n) = 1^2 + 1 + 1 = 3</p><p><strong>Step 4:</strong> Alternatively, if x → 1, then x^n → 1 when x = 1 exactly. But the functional equation gives x^2 + x + 1 evaluated at x = 1, which equals 3. The limiting behavior shows that lim(x^n) = 3.</p><p>∴ Answer: C (which is 3)</p>
Correct Answer: C