integration - Limit of an integral

I am trying to compute the following limit,

$$\lim_{k \to \infty} \int_0^1 \left| \frac{\sin(kx)}{x}\right| dx$$
After some numerical calculations, it loos like the limit is $\infty$. To prove it, I tried to use Riemann sums but such approach is not working.

Any hints on how one should prove this ?

Thanks in advance.

Answer

Changing variables ($y=kx$) this is

$$\lim_{k\to\infty}\int_0^k\frac{|\sin y|}{y}\,dy =\int_0^\infty\frac{|\sin y|}{y}\,dy.$$
This improper integral is well-known to diverge. For instance
$$\int_{(n-1)\pi}^{n\pi}\frac{|\sin y|}{y}\,dy \ge\frac{1}{n\pi}\int_0^\pi\sin y\,dy=\frac{2}{n\pi}.$$
Adding these up, and considering the behaviour of the harmonic

series shows the integral diverges.