Limit of two functions with expected values

I am tasked to prove the following limits. $X$ is a random variable and $\mathbb{E}[1/X] < \infty$ (or at least the latter can not be assumed, as this would make the proofs trivial).

$\lim_{y \rightarrow \infty} y \mathbb{E}\left[\frac{1}{X} \mathbf{1}_{\{X > y\}}\right] = 0$

$\lim_{y \rightarrow 0} y \mathbb{E}\left[\frac{1}{X} \mathbf{1}_{\{X > y\}}\right] = 0$

The approaches I've tried were fruitless. My first instinct was to find an upper bound for each expected value and prove that this goes to zero. The problem is that I don't know anything about the distribution of $X$, so it is not possible to bound the expected value integral this way. (Also, the upper limit of integration could be infinite, which might present issues.)

My second approach was to show that the limits of integration become the same in the first problem, and this suggests that the expected value is zero, e.g. after defining $Z = 1 / X$,

$\lim_{y \rightarrow \infty} \mathbb{E}\left[Z \mathbf{1}_{\{1 / Z > y\}}\right] = \lim_{y \rightarrow \infty} \int_{1/y}^0 z f_Z(z) dz = \int_0^0 z f_Z(z) dz = 0$

Unfortunately, that this causes the overall limit to equal zero is not clear:

$\lim_{y \rightarrow \infty} y \mathbb{E}\left[\frac{1}{X} \mathbf{1}_{\{X > y\}}\right] = \lim_{y \rightarrow \infty} y \cdot \lim_{y \rightarrow \infty} \mathbb{E}\left[\frac{1}{X} \mathbf{1}_{\{X > y\}}\right] = \infty \cdot 0 \neq 0$

This doesn't seem to go anywhere, and this approach does not work for the second limit I am tasked to prove.

Any helpful directions for this problem?

For every $y\gt0$, let $$X_y=\dfrac{y}X\mathbf 1_{X\gt y}.$$ Then $X_y\to0$ almost surely when $y\to0$ and when $y\to+\infty$ (can you check this?) and $|X_y|\leqslant1$ almost surely, for every $y$ (can you check this?), hence, by Lebesgue dominated convergence theorem, $E(X_y)\to0$ when $y\to0$ and when $y\to+\infty$.
To end this, note that, for every $y$, $$E(X_y)=yE\left(\dfrac1X\mathbf 1_{X\gt y}\right).$$