Asymptotes

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 behavior 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. 

Definition of Asymptote

An asymptote of a curve is the line formed by the movement of the curve and the 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, an 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.

The Application of an Asymptote in Real Life 

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.

A value that you grow closer to but never quite achieve is called an asymptote. A horizontal, vertical, or slanted line that a graph approaches but never touches is known as an asymptote in mathematics.

Assume you're returning to your automobile, which is parked at the far corner of the mall parking lot. You're fatigued after a long day of shopping. You're halfway to your car after about a minute of walking, but you notice you're slowing down. You've only walked a quarter of the way to your car after another minute.

Your legs are becoming very heavy. You're only 1/8 of the way thereafter the third minute. The pattern persists. It appears as if you would never get there as your shopping-tired body lags even further. Each additional minute only gets you halfway to the car compared to where you were before.

Well, here's the kicker: In principle, if this trend continues, you will never get there. Yes, you're getting closer to your car by the minute, but the distances you're covering are shrinking. With time, you get incredibly close. You get so near to your car that you can feel it. You get so near that your steps are as little as a pinhead, but you never cross more than half of the remaining distance to the car.

In actuality, this circumstance, known as Zeno's dilemma, is a bit silly: no matter how weary you are, you eventually walk more than half the distance to your car. You just have one step left to travel, so you simply take it and arrive; yet, mathematicians frequently prefer to consider it in terms of theory.

Types of Asymptotes

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. 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.

In fig.4a, you can find two horizontal asymptotes, in fig.4b, there are 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.

How to Find Asymptotes of a 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- 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 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 from (+) positive infinity to (-) negative infinity.

Essential Characteristics of Asymptotes

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 the 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.

What are Asymptotes and How can I Find Them?

The equation for an asymptote is x = a, y = a, or y = axe + b because it is a horizontal, vertical, or slanting line. The rules for finding all forms of asymptotes of a function y = f are as follows (x).

A horizontal asymptote has the form y = k, where x or x - is a positive or negative number. lim x f(x) and lim x - f(x) are the values of one or both of the limits (x). Click here to learn how to discover the horizontal asymptote using tricks and shortcuts.

A vertical asymptote has the form x = k, where y or y - is a positive or negative number. A slant asymptote has the form y = mx + b, where m is less than zero. An oblique asymptote is another term for a slant asymptote. It is commonly found in rational functions, and mx + b is the quotient obtained by dividing the numerator by the denominator of the rational function. In the next sections, we'll look at the process of locating each of these asymptotes in greater depth.

Finding a Rational Function's Horizontal Asymptotes

• The degrees of the polynomials in the numerator and denominator of the function are used to find the horizontal asymptote.

• Divide the coefficients of the leading phrases if both polynomials have the same degree. It will give you the value of the asymptote. 

• If the numerator's degree is smaller than the denominator, the asymptote is found at y = 0. (which is the x-axis).

• There is no horizontal asymptote if the numerator's degree is bigger than the denominator. 

Finding a Rational Function's Vertical Asymptotes

To locate the vertical asymptote of a rational function, reduce it to its simplest form, set the denominator to zero, then solve for x values.

Examples of Asymptotes

In the question, you will have to follow some steps to recognise the different types of asymptotes. 

1. 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 

Fun Facts About Asymptotes 

1. If the degree of the denominator is greater than the degree of the numerator, the 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.