Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$
Let's take the limitso,
equation of the horizontal asymptote on the left:
$$y = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$
$$\lim_{x \to \infty} \operatorname{sign}{\left(\sin{\left(x \right)} \right)} = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$
Let's take the limitso,
equation of the horizontal asymptote on the right:
$$y = \operatorname{sign}{\left(\left\langle -1, 1\right\rangle \right)}$$