A Ratio of Integrals

Problem

Suppose $C, L \in \mathbb{R}$ with $L \neq 0$ satisfy

\lim_{n \to \infty} \frac{n^C \displaystyle\int_0^{\pi/2} x^n \sin x \, dx}{\displaystyle\int_0^{\pi/2} x^n \cos x \, dx} = L.

Find $L$ .

Answer

Solution

Difficulty	9/10
Topics	Limits, calculus, Recursion

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