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.

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