**If the graph of a function y = f (x) goes out to infinity, each “end” of the graph may get closer and closer to a straight line, which is then called an asymptote.** In other words, some straight line may well approximate the graph as the function values of

*y*or the values of

*x*become very large. In calculus, asymptotes are traditionally classified as either horizontal, vertical, or slant.

If *y* = *f*(*x*) is a continuous, differentiable function for −∞ < *x* < +∞, then the horizontal line *y* = *L* is a horizontal asymptote if either Eq. (1*a*) or Eq. (1*b*) is true.

This means that the farther out *x* gets along the *x* axis, the closer the graph of *y* = *f* (*x*) gets to the line *y* = *L*. In particular, lim_{x→±∞}{*f*(*x*) − *L*} = 0 if *y* = *L* is an asymptote to the graph in both directions. In addition, the tangent lines to the graph of *y* = *f* (*x*) approach the horizontal asymptote as *x* → +∞ or *x* → −∞, as the case may be.

In contrast, if the function *y* = *f* (*x*) is not defined at the point *x* = *a*, the vertical line *x* = *a* is a vertical asymptote if either Eq. (2*a*) or Eq. (2*b*) is true.

Here, the notation *x* → *a*^{+} means that *x* is approaching *a* from its right, its positive side, and the notation *x* → *a*^{−} means that *x* is approaching the point *x* = *a* from its left, its negative side. In each case, the limit is either +∞ or −∞.

The **illustration** represents the graph of a hyperbola that we can assume to be either Eq. (3*a*) or Eq. (3*b*).

We see fromS Eq. (3*b*) that the line *y* = 3 is a horizontal asymptote for each end, *x* → +∞ or *x* → −∞, while the vertical line *x* = 1 is a vertical asymptote as x approaches 1 from the right or from the left. * See also: ***Hyperbola**

It is less common to look for slant asymptotes. A straight line, *y* = *mx* + *n*, is a slant asymptote for *y* = *f* (*x*) if Eq. (4) is true,

that is, if the distance between the graph *y* = *f* (*x*) and the line *y* = *mx* + *n* goes to 0 as *x* → +∞ or *x* → −∞. If *m* = 0, the asymptote is horizontal.

By dividing Eq. (4) by *x* and then taking *x* → +∞, we can obtain a formula for *m*. Using *m*, we can then find a formula for *n*. Sticking to the case *x* → +∞, the formulas are Eqs. (5*a*) and (5*b*).

We may consider the simple example given by Eq. (6).

Its graph has, in addition to a vertical asymptote, a slant asymptote. Using our formulas, we find first that *m* = 1 and then that *n* = 0. The slant asymptote is *y* = *x*. The same calculation works for both *x* → +∞ and *x* → −∞. Therefore, *y* = *x* is the slant asymptote for both “ends” of the graph. The slope of the tangent line to the graph at the point *x* is *f*′(*x*) = 1 − 1/*x*^{2}. When *x* → ±∞, this slope tends to 1, confirming our prior computation.

The terms asymptotic analysis and asymptotic expansion are more general terms used in the study of the behavior of a function, for example, the solution of a differential equation or the summation of an infinite series, for large values of its independent variables. Such studies are essential, especially in applied mathematics. To give a simple example, one might write Eq. (7). * See also: ***Analytic geometry**; **Calculus**; **Differential equation**; **Differentiation**; **Series**