Find the limit $\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$

Question

Problem Statement

Find the limit $$\lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx$$

Attempt

This problem is very similar to the following and I am basically going to work it out the same way. First, $\log(1-\frac{x}{n})\leq-\frac{x}{n}$ when $0\leq \frac{x}{n}\leq 1$. Therefore, on $[0,n]$, \begin{align*} \bigg\lvert\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n))\bigg\rvert&=\bigg\lvert e^{n\log\left(1-\frac{x}{n}\right)}\log(2+\cos(x/n))\bigg\rvert \\ &\leq \bigg\lvert e^{n(\frac{-x}{n})}\log(3)\bigg\rvert\\ &=e^{-x}\log(3). \end{align*} Since $e^{-x}\log(3)$ is integrable and non-negative, the dominated convergence tells us that \begin{align*} \lim_{n\to\infty} \int_0^n\left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n)) \ dx&=\int \lim_{n\to\infty} \left(1-\frac{x}{n}\right)^n\log(2+\cos(x/n))\chi_{[0,n]} \ dx\\ &=\int_0^{\infty}e^{-x}\log(3)\\ &=\log(3). \end{align*}

Is this the correct way to apply the dominated convergence theorem here?

Answer 1

As an alternative, since $\log(2+\cos x)$ is an analytic even function over $[-1,1]$,

$$\begin{eqnarray*} n\int_{0}^{1}(1-x)^n \log(2+\cos x)\,dx&=&\log 3\int_{0}^{1}n(1-x)^n\,dx+O\left(\int_{0}^{1}nx^2(1-x)^n\,dx\right)\\&=&\frac{n}{n+1}\,\log 3+O\left(\frac{1}{n^2}\right).\end{eqnarray*} $$