Find $ \lim_{n \to \infty} \int_{0}^{1} f(x) \sin(nx)dx$ where $f$ is continuously differentiable over $[0,1]$

Question

Start by using integration by parts, then use the fact that since $f'$ is continuous, there is a constant $M$ such that $M \geq |f'(x)| \forall x\in[0,1]$.

I used integration by parts and got:

$$ I=\lim_{n \to \infty} \int_{0}^{1} f(x) \sin(nx)dx=\lim_{n \to \infty} \left[-\dfrac{1}{n} \int_{0}^{1} f'(x) \cos(nx)dx \right] $$

Stuck now. No clue how to use that $M$ information with the squeeze theorem. My instinct would be to assume $M>f'(x)$ such that: $$ \int_{0}^{1} (-M) \cos(nx)dx \leq \int_{0}^{1} f'(x) \cos(nx)dx \leq \int_{0}^{1} (M) \cos(nx)dx $$

Not sure if I bounded the above correctly, but then I'd do:

$$ \left[ -\dfrac{M\sin(nx)}{n} \right]_{x=0}^{x=1} \leq \int_{0}^{1} f'(x) \cos(nx)dx \leq \left[ \dfrac{M\sin(nx)}{n} \right]_{x=0}^{x=1} $$ $$ -\dfrac{M\sin(n)}{n} \leq \int_{0}^{1} f'(x) \cos(nx)dx \leq \dfrac{M\sin(n)}{n} $$

Then I divide the whole thing by $n$, and take the limit as $n \to \infty$ such that the middle part of the inequality becomes $I$.

$$ -\lim_{n\to\infty}\dfrac{M\sin(n)}{n^2} \leq I \leq \lim_{n\to\infty}\dfrac{M\sin(n)}{n^2} $$

Evaluating the limits I get

$$ 0 \leq I \leq 0 $$

Thus, $I=0$. Not sure if this is the right approach.

This question has been answered, but using different methods out of the scope of this class. Can someone please solve it using the above hint.

Answer 1

Your attempt is pretty much it. Here is how I would write it down:

Integrating by parts,

$$\int_0^1 f(x) \sin(nx) dx = \frac{-1}{n}\int_0^1 f(x) d \cos(nx)$$ $$= \frac{-1}{n}f(x) \cos(nx)\big\vert_{x=0}^{x=1} + \frac{1}{n} \int_0^1 \cos(nx) f'(x) dx$$

Now, $$\frac{-1}{n}f(x) \cos(nx)\big\vert_{x=0}^{x=1}= \frac{-1}{n} f(1) \cos(n) + \frac{1}{n} f(0) \stackrel{n \to \infty}\longrightarrow 0$$

and $$\left|\frac{1}{n}\int_0^1 \cos(nx) f'(x) dx\right| \leq \frac{1}{n} \int_0^1 |f'(x)|dx \leq \frac{1}{n} \Vert f' \Vert_\infty \stackrel{n \to \infty}\longrightarrow 0$$ where we used that $f'$ is bounded since it is continuous on the compact interval $[0,1]$.

Combining all this, we see that $$\lim_{n\to \infty} \int_0^1 f(x) \sin(nx) dx = 0$$