Asymptote Formula

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Asymptotes

An asymptote of a curve y=f(x) that has an infinite branch is called a line such that the distance between the point(x, f(x) ) lying on the curve and the line approaches zero as the point moves along the branch to infinity.

Asymptotes can be vertical, oblique (slant) and horizontal. A horizontal asymptote is often considered as a special case of an oblique asymptote.

Vertical Asymptote

The straight line x=a is a vertical asymptote of the graph of the function y=f(x) if at least one of the following conditions is true:

In other words, at least one of the one-sided limits at the point x=a must be equal to infinity.

A vertical asymptote occurs in rational functions at the points when the denominator is zero and the numerator is not equal to zero (i.e. at the points of discontinuity of the second kind). For example, the graph of the function y=1/x has the vertical asymptote x=0 (Figure 1). In this case, both one-sided limits (from the left and from the right) tend to infinity:

A function which is continuous on the whole set of real numbers has no vertical asymptotes.

When x approaches some constant value c from left or right, the curve moves towards infinity (i.e.,∞)  , or -infinity (i.e., -∞) and this is called Vertical Asymptote.

Oblique Asymptote

The straight line y=kx+b is called an oblique (slant) asymptote of the graph of the function

Therefore, when finding oblique (or horizontal) asymptotes, it is a good practice to compute them separately.

The coefficients k and b of an oblique asymptote y=kx+b are defined by the following theorem:

When x moves towards infinity (i.e.,∞)  , or -infinity (i.e., -∞), the curve moves towards a line y = mx + b, called Oblique Asymptote.

Please note that m is not zero since that is a Horizontal Asymptote.

How to Find Horizontal Asymptotes?

To recall that an asymptote is a line that the graph of a function approaches but never touches. In the following example, a Rational function consists of asymptotes.

In the above example, we have a vertical asymptote at x = 3 and a horizontal asymptote at y = 1. The curves approach these asymptotes but never visit them.

The method to identify the horizontal asymptote changes based on how the degrees of the polynomial in the function’s numerator and denominator are compared. Below are the points to remember to find the horizontal asymptotes:

  • If the degree of the polynomials both in numerator and denominator is equal, then divide the coefficients of highest degree terms to get the horizontal asymptotes.
  • If the degree of the numerator is less than the degree of the denominator, then the horizontal asymptotes will be y = 0.
  • If the degree of the numerator is greater than the degree of the denominator, then there are no horizontal asymptotes.

Asymptotes of a Hyperbola

Hyperbola contains two asymptotes. Two bisecting lines that are passing by the center of the hyperbola that doesn’t touch the curve are known as the Asymptotes. These can be observed in the below figure.

If the centre of a hyperbola is (x0, y0), then the equation of asymptotes is given as:

If the centre of the hyperbola is located at the origin, then the pair of asymptotes is given as:

y = ±(b/a)x

That means,

y = (b/a)x

y = -(b/a)x

Let us see some examples to find horizontal asymptotes.

Proof

Necessity

Sufficiency

Suppose that there are finite limits

Horizontal Asymptote

When x moves to infinity or -infinity, the curve approaches some constant value b, and is called a Horizontal Asymptote.

Asymptotes of a Parametrically Defined Curve

Let a plane curve be defined by the parametric equations

Asymptote of a Polar Curve

Consider a curve given in polar coordinates by the equation

Its asymptote (if it exists) can be described by the two parameters: the distance p from the center to the asymptote (segment OA in Figure 3) and the angle ∝ of inclination of the asymptote to the polar axis.

Asymptotes of an Implicit Curve

An implicitly defined algebraic curve is described by the equation

The above formulas for the asymptotes of an implicit curve are valid if the curve has no singular points at infinity.

Rational functions and Asymptotes

This graph follows a horizontal line ( red in the diagram)  as it moves out of the system to the left or right. This is a horizontal asymptote with the equation y = 1. As x gets near to the values 1 and –1  the graph follows vertical lines ( blue). These vertical asymptotes occur when the denominator of the function, n(x),  is zero ( not the numerator).
To find the equations of the vertical asymptotes we have to solve the equation:

x2 – 1 =  0

         x2 = 1

          x = 1 or x = –1

Near to the values x = 1 and x = –1 the graph goes almost vertically up or down and the function tends to either +∞ or –∞.

We get a horizontal asymptote because the numerator and the denominator,  t(x) = x2 and n(x) = x2 – 1 are almost equal as x gets bigger and bigger.
If, for example, x = 100 then x= 10000 and x2 – 1 = 9999 , so that when we divide one by the othere we get almost 1. The bigger the value of x the nearer we get to 1.

Vertical asymptotes can be found by solving the equation n(x) = 0 where n(x) is the denominator of the function ( note: this only applies if the numerator t(x) is not zero for the same x value).

Horizontal asymptotes can be found by finding the limit

An asymptote is a straight line that constantly approaches a given curve but does not meet at any infinite distance. In other words, Asymptote is a line that a curve approaches as it moves towards infinity.

The curves visit these asymptotes but never overtake them. The method opted to find the horizontal asymptote changes involves comparing the degrees of the polynomials in the numerator and denominator of the function. If both the polynomials have the same degree, divide the coefficients of the largest degree terms.

Asymptotes Meaning

An asymptote of the curve y = f(x) or in the implicit form: f(x,y) = 0 is a straight line such that the distance between the curve and the straight line lends to zero when the points on the curve approach infinity.

There are three types of asymptotes namely:

  • Vertical Asymptotes
  • Horizontal Asymptotes
  • Oblique Asymptotes

The point to note is that the distance between the curve and the asymptote tends to be zero as it moves to infinity or -infinity.

How to Find the Equation of Asymptotes

In pre-calculus, you may need to find the equation of asymptotes to help you sketch the curves of a hyperbola. Because hyperbolas are formed by a curve where the difference of the distances between two points is constant, the curves behave differently than other conic sections. This figure compares the different conic sections.

Cutting the right cone with a plane to get conic sections.

Because distances can’t be negative, the graph has asymptotes that the curve can’t cross over.

Creating a rectangle to graph a hyperbola with asymptotes.

Solved Problems

Notice that the degree of the numerator is one more than the degree of the denominator. Therefore, the function has an oblique asymptote. We can find its equation by long division:

that is the asymptote (if it exists) is positioned horizontally. Determine the parameter p:

Here we took into account that the special trigonometric limit is equal to 1.

Thus, the cissoid of Diocles has the vertical asymptote x=2a (Figure 24)

This is what the calculator shows us. The graph actually crosses its asymptote at one point. (This can never happen with a vertical asymptote).

If we want to speculate on further possibilities we can see that if the degree of the numerator is 2 degrees greater than that of the denominator then the graph goes out of the coordinate system following a parabolic curve and so on.

Example 4

Find the asymptotes of the function

In this example the division has already been done so that we can see there is a slanting asymptote with the equation  y = x.

To find the vertical asymptotes we solve the equation  n(x) = 0.

   x2 – 1 = 0

         x2 = 1

          x = 1 or x = –1

The vertical asymptotes are x = 1 and x = –1.

Here’s the graph

Example 1:

Find the horizontal asymptotes for f(x) = x+1/2x

Solution:

Given, f(x) = (x+1)/2x

Since the highest degree here in both numerator and denominator is 1, therefore, we will consider here the coefficient of x.

Hence, horizontal asymptote is located at y = 1/2

Example 2:

Find the horizontal asymptotes for f(x) = x/x2+3

Solution:

Given, f(x) = x/x2+3

Since we can see here the degree of the numerator is less than the denominator, therefore, the horizontal asymptote is located at y = 0.

Example 3:

Find the horizontal asymptotes for f(x) =(x2+3)/x+1

Solution:

Given, f(x) =(x2+3)/x+1

As you can see, the degree of the numerator is greater than that of the denominator. Hence, there is no horizontal asymptote.

Identify vertical and horizontal asymptotes

ASYMPTOTES OF RATIONAL FUNCTIONS

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