Limits, Continuity & Differentiability
Limits
Grade None
Question:
<p>Let \(\text{Lt}_{n \to \infty} f(n) = l\). Given that \(\text{Lt}_{n \to \infty} f(n+1) = \dfrac{1}{2} \text{Lt}_{n \to \infty} \left[f(n) + \dfrac{9}{f(n)}\right]\) and \(f(n) > 0 \; \forall n \in N\), find \(\text{Lt}_{n \to \infty} f(n)\).</p>
Step-by-Step Solution
Key Concept: Since f(n) → l as n → ∞, both f(n+1) and f(n) approach the same limit l. Substitute this limiting behavior into the given functional equation to obtain an algebraic equation in l.
<p><strong>Step 1:</strong> Given that Lt<sub>n → ∞</sub> f(n) = l, we also have Lt<sub>n → ∞</sub> f(n+1) = l (since shifting index by 1 doesn't change the limiting behavior for n → ∞).</p><p><strong>Step 2:</strong> Taking the limit as n → ∞ on both sides of the given equation:</p><p>l = (1/2)[l + 9/l]</p><p><strong>Step 3:</strong> Multiply both sides by 2:</p><p>2l = l + 9/l</p><p><strong>Step 4:</strong> Simplify:</p><p>l = 9/l</p><p><strong>Step 5:</strong> Multiply both sides by l:</p><p>l² = 9</p><p><strong>Step 6:</strong> Since f(n) > 0 for all n ∈ ℕ, we have l > 0, therefore:</p><p>l = 3</p><p>∴ Answer: <strong>3</strong></p>
Correct Answer: 3