Final Answer:
The limit of (n+1)ᵏ - nᵏ / n⁽ᵏ⁻¹⁾ as n approaches infinity is k.
Step-by-step explanation:
To evaluate the limit, limₙ→∞ (n+1)ᵏ - nᵏ / n⁽ᵏ⁻¹⁾, we can use the fact that limₙ→∞ [(n+1) / n] = 1. Applying this, we simplify the expression by dividing both the numerator and denominator by nᵏ.
This simplification results in limₙ→∞ [(1 + 1/n)ᵏ - 1] / (n⁻¹/ⁿ). As n approaches infinity, the term 1/n becomes negligible, and the expression simplifies further to limₙ→∞ [(1)ᵏ - 1] / 0. Since (1)ᵏ remains 1 for any value of k, the limit becomes limₙ→∞ [1 - 1] / 0, which is an indeterminate form.
To resolve the indeterminate form, we can apply L'Hôpital's Rule by taking the derivative of the numerator and denominator with respect to n. After differentiation, the expression becomes limₙ→∞ [k/n] / (-n⁻²/ⁿ). Simplifying further, we get limₙ→∞ -k / n⁻¹, which equals 0 as n approaches infinity.
Therefore, the limit of (n+1)ᵏ - nᵏ / n⁽ᵏ⁻¹⁾ as n approaches infinity is k. This result illustrates the convergence of the expression to the value of k under the given conditions.