The (first part of the) problem reads:
Show that $$f_t(x)=\frac{\sin(xt)}{\pi x}\to\delta \ \text{in} \ \mathcal{D}'(\mathbb{R}) \ \text{as} \ t\to\infty.$$
If I understand correctly, this means that I am supposed to show that for each $\phi\in C_C^{\infty}(\mathbb{R})$ $$\lim_{t\to\infty}\int_{\mathbb{R}}\frac{\sin(xt)}{\pi x}\phi(x)\text{d}x=\phi(0)=\delta(\phi),$$ correct?
If so, any tips on how to calculate this integral (or otherwise, prove this result)?
The integral can be rewritten using a variable substitution: $$ \int \frac{\sin xt}{\pi x} \, \phi(x) \, dx = \frac{1}{\pi} \int \frac{\sin xt}{xt} \, \phi(x) \, t \, dx = \{ s = xt \} = \frac{1}{\pi} \int \frac{\sin s}{s} \, \phi(s/t) \, ds $$
Here, $\phi(s/t) \to \phi(0)$ as $t \to \infty$ and the integral $\int_{-\infty}^{\infty} \frac{\sin s}{s} \, dx = \pi,$ which suggests that $\frac{\sin xt}{\pi x}$ converges to $\delta(x).$ But you need to fill in the details.