Asymptote

An asymptote is a line or a curve that the graph of a function approaches, as shown in the figure below:



The asymptote is indicated by the vertical dotted red line, and is referred to as a vertical asymptote.

Types of asymptotes

There are three types of linear asymptotes.

Vertical asymptote

A function f has a vertical asymptote at some constant a if the function approaches infinity or negative infinity as x approaches a, or:

Referencing the graph below, there is a vertical asymptote at x = 2 since the graph approaches either positive or negative infinity as x approaches 2.



Horizontal asymptote

A function f has a horizontal asymptote at some constant a if the function approaches a as x approaches negative or positive infinity, or:

In the figure below, f(x) approaches y = 2 as x approaches either negative or positive infinity, so y = 2 is a horizontal asymptote.



In certain cases, it is possible for the graph of a function to intersect its horizontal asymptote, as shown in the figure below:



Furthermore, the graph of a function may have multiple horizontal and vertical asymptotes:



Referencing the figure above, f(x) has vertical asymptotes at x = -3, x = 2, and x = 5; it has a horizontal asymptote at y = 2.

Oblique asymptote

A function f has an oblique (slant) asymptote if it approaches a line of the form y = mx + b (where m ≠ 0) as x approaches negative or positive infinity. The graph of is shown in the figure below. It has an oblique asymptote at y = x - 1.



How to find the asymptotes of a rational function

A rational function is a function that can be written in the form . The techniques below apply to rational functions in which P(x) and Q(x) are polynomial functions and Q(x) ≠ 0.

Vertical asymptotes

To find the vertical asymptotes of a rational function f of the form described above:

Examples

Find any vertical asymptotes for the following functions:


i. The zeros of Q(x) occur when (x - 2) = 0 and (x + 3) = 0, so x = 2 and x = -3. Since there are no shared factors with P(x), f(x) has vertical asymptotes at x = 2 and x = -3, since these values of x result in f(x) being undefined. The graph of f(x) is shown in the figure below:



ii. Q(x) factors to (x + 2)(x - 2), so the zeros occur at x = ± 2, and f(x) is undefined at x = ±2. Since P(x) and Q(x) share the factor (x - 2), f(x) can be simplified as follows:

Thus, f(x) has a hole at x = 2, and a vertical asymptote at x = -2, as shown in the figure below.



Horizontal asymptotes

To find a horizontal asymptote for a rational function of the form , where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0:

Examples

Find any horizontal asymptotes for the following functions:


i. The degree of Q(x) is 4, since the highest order term of q(x) is x4. Similarly, the degree of P(x) is 3. Since Q(x) > P(x), f(x) has a horizontal asymptote at y = 0, as shown in the figure below.



ii. The degree of both P(x) and Q(x) are 3. The corresponding terms are 4x3 and 2x3, respectively, and their coefficients are 4 and 2. Thus, f(x) has a horizontal asymptote at y = 4/2 = 2, as shown in the figure below:



Oblique asymptotes

To find an oblique asymptote for a rational function of the form , where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0:

Example

Find any slant asymptotes for

Since the degree of P(x) is exactly one greater than Q(x), f(x) has an oblique asymptote, and we divide Q(x) into P(x):



The quotient is s = x + 2, so f(x) has an oblique asymptote at y = x + 2, as shown in the figure below: