Horizontal asymptotes are the graphical manifestation of limits as x approaches infinity. Vertical asymptotes are the graphical manifestation of limits equal to infinity (at a finite x-value).
Thus, since . The graph will show a horizontal asymptote at y = 1.
Since the graph of approaches the x-axis as an asymptote, it follows that . (The fact that this graph crosses the x-axis many times on its trip to infinity is not a concern; the axis is still an asymptote.)
Since , the functions all have a vertical asymptote of x = 0.