Solution of $ \lim_{n\to\infty} \int_0^1 \sin^n(x)\sin\left(\dfrac{1}{x^n} \right) \,dx $

Question

I have to calculate the solution of $ \lim_{n\to\infty} \int_0^1 \sin^n(x)\sin\left(\dfrac{1}{x^n} \right) \,dx $. $\\$

My idea is to use Lebesgue's dominated convergence theorem $\\$.

For that, we have $f_n :[0,1] \rightarrow \mathbb{R}$ $x \mapsto \sin^n(x) \sin\left(\dfrac{1}{x^n}\right)$. $\\$ We can say that $|f_n(x)| \leq \sup f_n(x) \\$. As $\sup f_n$ is constant, it is integrable. $\\$

I found out by plotting the function that $\lim_{n\to\infty} f_n(x)= 0$ . $\\$

But how can I show that the limit is zero? With this information I could say that the integral converges to zero and the task woul be complete.
Thanks!

Answer 1

$\vert\sin^n(x)\sin(1/x^n)\vert\le\vert\sin(x)\vert^n$ and $\vert\sin(x)\vert<1$ for all $x\in[0,1]$.