Compute the limit $ \lim_{x\to \infty}\frac{1}{x}\int_0^x\frac{1}{2+\cos t}dt$

Question

Compute the following limit: $$ \lim_{x\to \infty}\frac{1}{x}\int_0^x\frac{1}{2+\cos t}dt$$ We have that $0\leq\frac{1}{2+\cos t}\leq1, \forall t\in \mathbb{R}$ which means that $0\leq\int_0^x\frac{1}{2+\cos t}dt\leq x$, so $0\leq\frac{1}{x}\int_0^x\frac{1}{2+\cos t}dt\leq 1$. The given limit is in $[0,1]$, but I don't think it helps evaluating it.

Answer 1

Hint

The function: $$f(t)=\frac{1}{2+\cos(t)}$$ as $2 \pi$ as a period.

With $n$ such that $2n \pi \leq x < 2(n+1) \pi$ (i.e $n=\left\lfloor\frac{x}{2\pi} \right\rfloor$) you have: $$\int_0^x f(x) dx=\int_0^{2n \pi} f(t) dt +\int_{2 n \pi}^x f(t)dt$$ but with linear change of variables and using periodicity: $$\int_0^{2n \pi} f(t) dt = \sum_{k=0}^{n-1} \int_{2k \pi}^{2(k+1) \pi} f(t) dt=\sum_{k=0}^{n-1} \int_{0}^{2( \pi} f(t) dt$$ so: $$\frac{1}{x} \int_0^x f(x) dx=\frac{\left\lfloor\frac{x}{2\pi} \right\rfloor}{x} \int_0^{2 \pi} f(t) dt+\frac{1}{x} \int_{\left\lfloor\frac{x}{2\pi} \right\rfloor}^x f(t) dt$$ it remains to prove that: $$\lim_{x \to \infty} \frac{\left\lfloor\frac{x}{2\pi} \right\rfloor}{x} =\frac{1}{2 \pi}$$ $$\lim_{x \to \infty} \frac{1}{x} \int_{\left\lfloor\frac{x}{2\pi} \right\rfloor}^x f(t) dt=0$$ to prove that the limit is:

$$\frac{1}{2 \pi} \int_0^{2 \pi} f(t) dt$$

Answer 2

For large $x$, the integral is a sum of whole periods and a residue. When averaging, only a single period remains:

$$\frac1{nT+\alpha T}\int_0^{nT+\alpha T}=\frac{nI+\beta I}{nT+\alpha T}\to \frac IT$$ where $I$ is the integral over a period, and $\alpha,\beta$ are less than one.

Using WA, the average over a period is $\dfrac1{\sqrt 3}$.

Answer 3

Hint: For the integral one can solve using $u\mapsto \tan\left(\frac{t}{2}\right)$, which gives

$$\begin{align*} \int \frac{2du}{(u^2+1)((1-u^2)(u^2+1)^{-1} + 2)} &= 2 \int \frac{du}{u^2+3} \\ &= \frac{2\tan^{-1}\left(\frac{u}{\sqrt3}\right)}{\sqrt3} + c \\ &= \frac{2}{\sqrt3} \tan^{-1}\left( \frac{\tan(t/2)}{\sqrt3} \right) + c \end{align*} $$

Evaluate this from $0$ to $x$ and use in the limit.