I have been working on this limit for days, but I am not getting it. The question is
Compute the limit $$\lim_{t \to + \infty} \int_0^{+ \infty} \frac{ \mathrm d x}{e^x+ \sin (tx)}$$
Note that the integral is well defined and convergent for every $t >0$. Indeed the integrand function is a positive function for every $t >0$ since
$$e^x + \sin tx > e^x-1 > x>0$$
And as $x \to + \infty$ the integrand function behaves like $e^{-x}$.
WHAT I TRIED:
I consider $t=2n \pi$ a multiple of $2 \pi$, and see what happens:
$$\int_0^{+ \infty} \frac{ \mathrm d x}{e^x+ \sin (2n \pi x)} = \sum_{k=0}^\infty \int_{k /n}^{(k+1) /n} \frac{ \mathrm d x}{e^x+ \sin (2n \pi x)}$$
Making the change of variables $u = 2n \pi x$ I get
\begin{align}\sum_{k=0}^\infty \frac{1}{2n \pi} \int_{2k \pi}^{(2k+2) \pi} \frac{ \mathrm d u}{e^{u/2n \pi}+ \sin (u)} &\ge
\sum_{k=0}^\infty \frac{1}{2n \pi} \int_{2k \pi}^{(2k+2) \pi} \frac{ \mathrm d u}{e^{(2k+2) \pi/2n \pi}+ \sin (u)} \\&=
\sum_{k=0}^\infty \frac{1}{2n \pi} \int_{2k \pi}^{(2k+2) \pi} \frac{ \mathrm d u}{e^{(k+1)/n}+ \sin (u)}\end{align} where I write the lower bound with the minimum of the function at $u=(2k+2) \pi$. Now I use the fact that the integrand function does is integrated over a period of $2 \pi$, and using the result for $C>1$
$$\int_0^{2 \pi} \frac{ \mathrm d u}{C+ \sin (u)} = \frac{2 \pi}{\sqrt{C^2-1}}$$
I get the estimate
\begin{align}\sum_{k=0}^\infty \frac{1}{2n \pi} \int_{2k \pi}^{(2k+2) \pi} \frac{ \mathrm d u}{e^{(k+1)/n}+ \sin (u)} &= \sum_{k=0}^\infty \frac{1}{2n \pi} \frac{2 \pi}{\sqrt{e^{2(k+1)/n} -1 }} \\&= \frac{1}{n} \sum_{k=0}^\infty \frac{1}{\sqrt{e^{2(k+1)/n} -1 }}\end{align}
Summing all up, I got that
$$\int_0^{+ \infty} \frac{ \mathrm d x}{e^x+ \sin (2n \pi x)} \ge \frac{1}{n} \sum_{k=0}^\infty \frac{1}{\sqrt{e^{2(k+1)/n} -1 }}$$
As $n \to \infty$ the series converges to the Riemann integral
$$\int_0^{+ \infty} \frac{\mathrm d y}{\sqrt{e^{2y}-1}} = \frac{\pi}{2}$$
Hence the limit should be a number larger than $\pi/2$, or $+ \infty$.
Using WA I got for large values of $t$ that the integral is between $1$ and $2$, thus $\pi/2$ could be the actual limit.
Best Answer
A slight modification of OP's attempt will lead to a solution. Indeed, write $I(t)$ for the integral and note that
$$ I(t) = \int_{0}^{\infty} \frac{\mathrm{d}x}{e^x + \sin(tx)} \stackrel{(y=tx)}= \frac{1}{t} \int_{0}^{\infty} \frac{\mathrm{d}y}{e^{y/t} + \sin y}. $$
Also, define $J(t)$ by
$$ J(t) = \frac{1}{t} \sum_{k=1}^{\infty} \int_{0}^{2\pi} \frac{\mathrm{d}y}{e^{2\pi k/t} + \sin y} = \sum_{k=1}^{\infty} \frac{2\pi/t}{\sqrt{e^{4\pi k/t} - 1}}, $$
where the second step follows from the integration formula
$$ \int_{0}^{2\pi} \frac{\mathrm{d}y}{c + \sin y} = \frac{2\pi}{\sqrt{c^2 - 1}}, \qquad c > 1. \tag{1} $$
Then similarly as in OP's attempt, we obtain
$$ J(t) \leq I(t) \leq J(t) + \frac{1}{t} \int_{0}^{2\pi} \frac{\mathrm{d}y}{e^{y/t} + \sin y}. \tag{2} $$
Now we observe:
Since the map $ u \mapsto \frac{1}{\sqrt{e^{2u} - 1}} $ is monotone decreasing, we have $$ \int_{2\pi/t}^{\infty} \frac{\mathrm{d}u}{\sqrt{e^{2u} - 1}} \leq J(t) \leq \int_{0}^{\infty} \frac{\mathrm{d}u}{\sqrt{e^{2u} - 1}}. $$ So by the squeezing theorem, we get $$ \lim_{t \to \infty} J(t) = \int_{0}^{\infty} \frac{\mathrm{d}u}{\sqrt{e^{2u} - 1}} = \frac{\pi}{2}. $$
Observe that $$ \int_{\pi}^{2\pi} \frac{\mathrm{d}y}{c + \sin y} \stackrel{\text{(1)}}\leq \frac{2\pi}{\sqrt{c^2 - 1}}. $$ From this, we have \begin{align*} \frac{1}{t} \int_{0}^{2\pi} \frac{\mathrm{d}y}{e^{y/t} + \sin y} &\leq \frac{1}{t} \left( \int_{0}^{\pi} \mathrm{d}y + \int_{\pi}^{2\pi} \frac{\mathrm{d}y}{e^{\pi/t} + \sin y} \right) \\ &\leq \frac{1}{t} \left( \pi + \frac{2\pi}{\sqrt{e^{2\pi/t} - 1}} \right). \end{align*} It is not hard to check that this bound converges to $0$ as $t \to \infty$.
Combining altogether and applying the squeezing theorem to $\text{(2)}$, we get
$$ \lim_{t\to\infty} I(t) = \frac{\pi}{2}. $$
Further Discussion:
Addendum. Regarding the extra question, the following heuristic approach gives a guess on the value of the constant $c$ in the asymptotic expansion $\text{(3)}$:
Note that, for $x > 0$ and $\theta \in \mathbb{R}$,
\begin{align*} \frac{1}{e^x + \sin\theta} &= \frac{1}{\sqrt{e^{2x}-1}} \biggl( 1 + 2 \sum_{k=1}^{\infty} \frac{(-1)^k}{(e^x + \sqrt{e^{2x}-1})^{2k-1}} \sin((2k-1)\theta) \\ &\hspace{7em} + 2 \sum_{k=1}^{\infty} \frac{(-1)^k}{(e^x + \sqrt{e^{2x}-1})^{2k}} \cos (2k\theta) \biggr). \end{align*}
Using this and substituting $\epsilon = 1/t$, we have
\begin{align*} I(t) &= \int_{0}^{\infty} \frac{\mathrm{d}x}{\sqrt{e^{2x}-1}} \\ &\quad + 2 \sum_{k=1}^{\infty} (-1)^k \int_{0}^{\infty} \frac{\sin((2k-1)x/\epsilon)}{\sqrt{e^{2x}-1}(e^x + \sqrt{e^{2x}-1})^{2k-1}} \, \mathrm{d}x \\ &\quad +2 \sum_{k=1}^{\infty} (-1)^k \int_{0}^{\infty} \frac{\cos(2kx/\epsilon)}{\sqrt{e^{2x}-1}(e^x + \sqrt{e^{2x}-1})^{2k}} \, \mathrm{d}x \\ &= \frac{\pi}{2} + 2 \sqrt{\epsilon} \sum_{k=1}^{\infty} (-1)^k \int_{0}^{\infty} \sqrt{\frac{\epsilon}{e^{2\epsilon u} - 1}} \frac{\sin((2k-1)u)}{(e^{\epsilon u} + \sqrt{e^{2\epsilon u}-1})^{2k-1}} \, \mathrm{d}u \\ &\hspace{3em} + 2 \sqrt{\epsilon} \sum_{k=1}^{\infty} (-1)^k \int_{0}^{\infty} \sqrt{\frac{\epsilon}{e^{2\epsilon u} - 1}} \frac{\cos(2ku)}{(e^{\epsilon u} + \sqrt{e^{2\epsilon u}-1})^{2k-1}} \, \mathrm{d}u, \end{align*}
where we utilized the substitution $x = \epsilon u$ in the last step. So it is reasonable to expect that $c$ in $\text{(3)}$ is given by:
\begin{align*} c &= 2 \sum_{k=1}^{\infty} (-1)^k \int_{0}^{\infty} \lim_{\epsilon \to 0^+} \sqrt{\frac{\epsilon}{e^{2\epsilon u} - 1}} \frac{\sin((2k-1)u)}{(e^{\epsilon u} + \sqrt{e^{2\epsilon u}-1})^{2k-1}} \, \mathrm{d}u \\ &\quad + 2 \sum_{k=1}^{\infty} (-1)^k \int_{0}^{\infty} \lim_{\epsilon \to 0^+} \sqrt{\frac{\epsilon}{e^{2\epsilon u} - 1}} \frac{\cos(2ku)}{(e^{\epsilon u} + \sqrt{e^{2\epsilon u}-1})^{2k-1}} \, \mathrm{d}u \\ &= 2 \sum_{k=1}^{\infty} (-1)^k \biggl( \int_{0}^{\infty} \frac{\sin((2k-1)u)}{\sqrt{2u}} \, \mathrm{d}u + \int_{0}^{\infty} \frac{\cos(2ku)}{\sqrt{2u}} \, \mathrm{d}u \biggr) \\ &= \sum_{k=1}^{\infty} (-1)^k \biggl( \sqrt{\frac{\pi}{2k-1}} + \sqrt{\frac{\pi}{2k}} \biggr), \end{align*}
where we utilized the identity
$$ \int_{0}^{\infty} \frac{\sin(a u)}{\sqrt{u}} \, \mathrm{d}u = \int_{0}^{\infty} \frac{\cos(a u)}{\sqrt{u}} \, \mathrm{d}u = \sqrt{\frac{\pi}{2a}}, \qquad a > 0. $$
Of course, interchanging the order of limit operators requires a great deal of care, especially in the situation like this where the absolute convergence fails. So this is not a proof yet, but rather a hand-waving heuristics. (Even the possibility that this guess is not true at all is still open to question!)