asymptote. For example, for the function has a limit of +∞ as , ƒ(x) has the vertical asymptote , even though ƒ(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or a vertical line in general) in more than one point. Moreover, if a function is continuous at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote. A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero. If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is at . This function has a vertical asymptote at because and . The derivative of is the function . For the sequence of points for that approaches both from the left and from the right, the values are constantly . Therefore, both one-sided limits of at can be neither nor . Hence doesn't have a vertical asymptote at . Horizontal asymptotes Horizontal asymptotes are horizontal lines that the graph of the function approaches as . The horizontal line y = c is a horizontal asymptote of the function y = ƒ(x) if or . In the first case, ƒ(x) has y = c as asymptote when x tends to , and in the second ƒ(x) has y = c as an asymptote as x tends to . For example, the arctangent function satisfies and So the line is a horizontal asymptote for the arctangent when x tends to , and is a horizontal asymptote for the arctangent when x tends to . Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function has a horizontal asymptote at y = 0 when x tends both to and because, respectively, Other common functions that have one or two horizontal asymptotes include (that has an hyperbola as it graph), the Gaussian function the error function, and the logistic function. Oblique asymptotes When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant