I would like to ask you for an explained solution to this problem:
Find the limit of the integral:
$$\lim_{n\to\infty}\int_{R_+}\frac{(x^n+1)\sin(x^2e^{-x^{2n}}+x^{-1}e^{-x^{-2n}})}{x^{n+2}+x^{-1}}dx $$
Use Lebesgue theorem for convergence.
Therefore I split the integral into two parts - on two intervals, as it has been also suggested in the problem:
$$ \int_{R_+}\frac{(x^n+1)\sin(x^2e^{-x^{2n}}+x^{-1}e^{-x^{-2n}})}{x^{n+2}+x^{-1}}dx = \int_0^1\frac{(x^n+1)\sin(x^2e^{-x^{2n}}+x^{-1}e^{-x^{-2n}})}{x^{n+2}+x^{-1}}dx + \int_1^{\infty}\frac{(x^n+1)\sin(x^2e^{-x^{2n}}+x^{-1}e^{-x^{-2n}})}{x^{n+2}+x^{-1}}dx$$
As I understand the theorem, I have to find the sequence of functions $f_n$ which is pointwise convergent to the function $f$ and each of these function is less than a certain integrable function $g(x)~~~(f_n(x)<g(x))$. This domination applies for all $n$ and $x$ in the domain $D$. Therefore the $f$ is integrable and:
$$ \lim_{n\to\infty}\int_D f_nd\mu = \int_D f $$
That's the theory. However, how to use this in this integral? I would appreciate your help in both solving and understanding.
EDIT
I have updated my knowledge on Lesbegue, however, I have still problems with pointwise convergence.
Let us simplify: $$\frac{(x^n+1)}{x^{n+2}+x^{-1}} = \frac{x^n(1+\frac{1}{x^n})}{x^n(x^2+\frac{1}{x^{n+1}})} = \frac{1+\frac{1}{x^n}}{x^2+\frac{1}{x^{n+1}}} $$
For $x\in(0,1)$ it tends to $\infty$ for $n \to \infty$; for $x\in(1,\infty)$, it tends to $\frac{1}{x^2}$.
However, I have still problems with $\sin(x^2e^{-x^{2n}}+x^{-1}e^{-x^{-2n}})$. I have assumed that:
$$|\sin(x^2e^{-x^{2n}}+x^{-1}e^{-x^{-2n}})|\leq x^2e^{-x^{2n}}+x^{-1}e^{-x^{-2n}}$$
$$ t=x^{2n}\\\sqrt[n]{t}e^{-t}+\frac{1}{\sqrt[2n]{t}}e^{-\frac{1}{t}} = \frac{\sqrt[n]{t}}{e^t}+\frac{1}{\sqrt[2n]{t}\sqrt[t]{e}}$$
For me, with $n \to \infty$ for $t\in(1,\infty)$ it tends to $1$ - and to $\infty$ for $t\in(0,1)$. Therefore it seems for me that there is a limit only on the interval $(1,\infty)$. However, I suppose that I have done something wrong, miscalculated or misestimated - please, point out my mistakes. I would ask you for some guidance, too.