Application of dominated convergence theorem- find limit

Question

Find (with justification) $$ \lim_{n\to \infty} \int_0^n (1+x/n)^{-n}\log(2+\cos(x/n))\,dx $$

Answer 1

Since $0<x<n$ it follows that $0<\dfrac{x}{n}<1$ and hence $\log(2+\cos(\dfrac{x}{n}))\leq\log3$.

Now $(1+\dfrac{x}{n})^{-n}=(1+\dfrac{x}{n})^{(n/x)(-x)}\leq2^{-x}$

So your dominating function is $3^{-x}\log 3$.

Now $(1+x/n)^{-n}\log(2+\cos(x/n))\to e^{-x}\log3$ so by Dominated Convergence Theorem,the integral converges to $\log3$.

Answer 2

Hint. You may observe that $$ \left(1+\frac x n \right)^{-n}\leq e^{-x}, \quad x\geq0, n \in \mathbb{N}^*, $$ and $$ \log (2+ \cos (x/n)) \leq \log (2+ |\cos (x/n)|) \leq \log (2+1)=\log 3 $$ thus $$ \mathbf{1}_{\large [0,n]}(x)\:\left(1+\frac x n \right)^{-n}\log (2+ \cos (x/n))\:\leq \mathbf{1}_{\large [0,\infty)}(x)\: e^{-x} \:\log 3 $$ and then use the dominated convergence theorem to conclude.