Evaluate $\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x/n)}{\sqrt{x+\cos(x/n)}}dx$

Question

The following problem came up on a qualifying exam.

Evaluate $\lim_{n \to \infty} \int_{0}^{n} \frac{\cos(x/n)}{\sqrt{x+\cos(x/n)}}dx$.

I was thinking that I'd try to bound the integrand and then to apply DCT but I cannot find a way to bound it by an $L^{1}((0,\infty))$ function. I am starting to believe it does not converge.

Answer 1

Use the substitution $x = ny$ to get

$$\lim_{n\to\infty}\int_0^1\frac{n\cos y}{\sqrt{ny+\cos y}}\:dy = \lim_{n\to\infty}\sqrt{n}\int_0^1\frac{\cos y}{\sqrt{y+\frac{\cos y}{n}}}\:dy$$

Taking just the integral portion into account, we have that on $(0,1)$

$$\frac{\cos y}{\sqrt{y+\frac{\cos y}{n}}} \leq \frac{1}{\sqrt{y}} \in L^1((0,1))$$

thus dominated convergence tells that

$$\lim_{n\to\infty}\int_0^1\frac{\cos y}{\sqrt{y+\frac{\cos y}{n}}}\:dy \to \int_0^1 2\cos(t^2)\:dt > 0$$

an integral of a continuous function on a compact interval (converges) which means

$$\lim_{n\to\infty}\sqrt{n}\int_0^1\frac{\cos y}{\sqrt{y+\frac{\cos y}{n}}}\:dy \to \infty$$

by properties of products of sequences.

Answer 2

Observe that

$$\int_{0}^{n} \frac{\cos(x/n)}{\sqrt{x+\cos(x/n)}}dx \ge \int_{0}^{n} \frac{\cos1}{\sqrt{x+1}}dx$$ $$ = 2\cos 1\sqrt{x+1}\,\big |_0^n = 2\cos 1(\sqrt{n+1}-1) \to \infty.$$