The asymptote of a curve is an important topic in the subject of Mathematics. It is part of analytic geometry. In simple words, asymptotes are in use to convey the behaviour and tendencies of curves. When the graph comes close to the vertical asymptote, it curves upward/downward very steeply. This way, even the steep curve almost resembles a straight line. It helps to determine the asymptotes of a function and is an essential step in sketching its graph. We also analyze how to find asymptotes of a curve. The detailed study of asymptotes of functions forms a crucial part of asymptotic analysis.

An asymptote of a curve is the line formed by the movement of curve and line moving continuously towards zero. This can happen when either the x-axis (horizontal axis) or y-axis (vertical axis) tends to infinity. In other words, Asymptote is a line that a curve approaches (without a meeting) as it moves towards infinity.

As you can see from the above illustrations, an asymptote of a curve is a line to which the curve converges. There is a peculiar and unique relationship between the curve and its asymptote. They run parallel to each other, but they never meet each other, at any point in infinity. They run very close to each other but are still apart.

Asymptotes have several applications, such as:

They are in use for significant O notations.

They are simple approximations for complex equations.

They are useful for graphing rational equations.

They are relevant for- Algebra: Rational functions and Calculus: Limits of functions.

As you may have noticed in Fig.1 and Fig. 2 above, sometimes a graph or the bend of the curve gets close enough to a line without ever touching it. This line is called an asymptote. Now, it is essential to know that an asymptote can be horizontal, vertical, or oblique/slanted.

Asymptotes are usually straight lines unless stated otherwise. You can even call an asymptote a value that you get closer to but never reach. In maths, as mentioned earlier, asymptotes can be horizontal, vertical, or an oblique/slanted line that a graph Approaches, but never touches it. Take a look at the illustration depicted in Fig.3 below to have a better understanding of the different types of asymptote (s).

As you have seen, there are three types of curves - horizontal, vertical, and oblique. It is important to note that the directions can also be negative. The curve can take an approach from any side, such as from above or below for a horizontal asymptote. Sometimes, and many times, a curve may even cross over, and move away and back again. Look at illustrations in Fig.4 below.

In fig.4a, you can find two horizontal asymptotes, in fig.4b, there two vertical asymptotes, and in fig.4c you can note that there are two oblique asymptotes. So, these figures explain the character of the curve and the lines (asymptotes) that run parallel to the curve.

The asymptote (s) of a curve can be obtained by taking the limit of a value where the function does not get a definition or is not defined. An example would be \infty∞ and -\infty −∞ or the point where the denominator of a rational function is zero.

Now you know that the curves walk alongside the asymptotes but never overtake them. The method in use to find horizontal asymptote changes- it is based on how the degree of the polynomials in the numerator and the denominator of the functions get a comparison. If the polynomials are equal in the degree, you can divide the coefficients of the largest degree values.

The vital point to note is that the distance between the curve and the asymptote tends to be zero when it moves form (+) positive infinity to (-) negative infinity.

In calculus, based on the orientation, curves of the form y = f(x) can be calculated using limits and can be any of the three forms

Horizontal Asymptotes - x goes to +infinity or –infinity, the curve approaches some constant value b. In curves in the graph of a function y = ƒ(x), horizontal asymptotes are flat lines parallel to x-axis that the graph of the function approaches as x moves closer towards +∞ or −∞.

Vertical Asymptote - when x approaches any constant value c, parallel to the y-axis, then the curve goes towards +infinity or – infinity.

Oblique Asymptote - when x goes to +infinity or –infinity, then the curve goes towards a line y=mx+b

On the question, you will have to follow some steps to recognise the different types of asymptotes. 1 - For example, find the domain and all asymptotes of the following function:

Y= x2 +3x +1

4x2 - 9

Solution = 4x2 – 9 = 0 (take denominator as zero)

x2 = 9/4 = 3/2

Y = x2 / 4x2

= 1/4

Domain x ≠ 3/2 or -3/2, Vertical asymptote is x = 3/2, -3/2, Horizontal asymptote is y = 1/4, and Oblique/Slant asymptote = none

2 – Find horizontal asymptote for f(x) = x/ x2+3

Solution= f(x) = x/ x2+3. As you can see, the degree of numerator is less than the denominator, hence, horizontal asymptote is at y= 0

1. If the degree of the denominator is greater than the degree of the numerator, horizontal asymptote is at y= 0.

2. If the degree of the denominator is less than the degree of the numerator by one, we get oblique asymptote

3. If the degree of the numerator is equal to the degree of the denominator, horizontal asymptote at a ratio of leading coefficients.

FAQ (Frequently Asked Questions)

Q1. How to Identify Oblique Asymptotes of Rational Functions?

The graph of function y=f(x) is oblique asymptote has a non-zero but finite slope. The graph approaches this point as x moves closer to +∞ or −∞ If the rational function has a fixed difference between numerator and denominator, then it can be termed as an oblique asymptote. Its numerator needs to be exactly one degree more than the denominator. After diving these two entities, the asymptote takes on the form of a polynomial term, as there will be a reminder as well as a linear term after the division.

Q2. What are the Rules for Vertical Asymptotes?

The most common form of linear function studies in calculus is y = f(x). If a real number k is a zero in evaluating the denominator Q(a) of a rational function, then there are two things to note.

f(a) = P(a) / Q(a) (i.e. Q(a) is the denominator)

P(a) and Q(a) have no common factors

The graph of this function has the vertical asymptote a=k

To arrive at the vertical asymptotes of a rational function, you need to set the denominator equal to zero and get the solution. Vertical asymptotes occur where the denominator is zero. Bear in mind that division by zero is not allowed.