Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi}\right) = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$
Tomamos como el límitees decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$
$$\lim_{x \to \infty}\left(\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi}\right) = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$
Tomamos como el límitees decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$