asymptote. A function ƒ(x) is asymptotic to the straight line (m ≠ 0) if In the first case the line is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case the line is an oblique asymptote of ƒ(x) when x tends to −∞. An example is ƒ(x) = x + 1/x, which has the oblique asymptote y = x (that is m = 1, n = 0) as seen in the limits Elementary methods for identifying asymptotes The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits). General computation of oblique asymptotes for functions The oblique asymptote, for the function f(x), will be given by the equation y = mx + n. The value for m is computed first and is given by where a is either or depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction. Having m then the value for n can be computed by where a should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining m exist. Otherwise is the oblique asymptote of ƒ(x) as x tends to a. For example, the function has and then so that is the asymptote of ƒ(x) when x tends to +∞. The function has and then , which does not exist. So does not have an asymptote when x tends to +∞. Asymptotes for rational functions A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes. The degree of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator. The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at x = 0, and x = 1, but not at x = 2. Oblique asymptotes of rational functions When the numerator of a rational function has degree exactly one greater