Asymptote to the Horizontal: Some Ground Regulations
As the name implies, a function's horizontal asymptote is a horizontal line that the function's graph "appears to coincide with," but which it does not. How a function will behave in the long run can be found by looking at its horizontal asymptote.
First, let's get a better understanding of the horizontal asymptote and the rules for locating it for various functions.
Horizontal Asymptote: What Is It?
If either lim x f(x) = k or lim x - f(x) = k, then the line y = k is the horizontal asymptote of the function y = f(x). i e meaning that as x or x -, the graph (curve) of the function appears to approach a line. It's more commonly known by the acronym HA. When x is very large or very small, the function tends to approach the real number k. A horizontal asymptote is not required for a function to exist. However, a function can only have a maximum of 2 asymptotes. i e One or two asymptotes may exist for a given function. Take a look at these illustrations to get a feel for the appearance of horizontal asymptotes. In most cases, a dotted line indicates a horizontal line. When the HA is the x-axis itself, the dotted line is typically not used.
To show the value that the function is tending toward, we find the HA while plotting a curve. However, it is not necessary to include it when drawing the curve on a graph. It is not possible to plot a horizontal asymptote on a graphing calculator because no such line exists. A horizontal line, in other words, does not exist in reality.
What About a Horizontal Asymptote? Can It Go Through the Bend?
The asymptote y = k of the function y = f(x) can indeed pass through the origin of the graph. i e for some values of x, f(x) = k Take heed that this is NOT the case with any vertical asymptote, as such an asymptote will NEVER touch the curve itself. Here is a case where the curve is intersected by the horizontal asymptote (HA).
In this case, the x-axis (whose equation is y = 0) crosses the curve at the origin (0, 0). The result of plugging zero into the equation x/(x2 1) = 0 is zero.
Find the Asymptote on the Horizon
How to Determine the Asymptote of a Horizontal Function y = f(x)
- The first thing to do is to determine the value of lim x f(x). i e The limit of the function can be calculated as x
- To proceed to Step 2, locate lim x - f(x). i e by using the limit formula [x] = [-x]
- Third, if both upper and lower bounds are real numbers, then the horizontal asymptote can be represented by y = k, where k is the limit value.
The answers or - in either of the two preceding cases are NOT the horizontal asymptotes and should be disregarded. It is possible that all of the bounds will result in the same value, in which case we would have just one HA (as in the following example). Learn more about boundary analysis by visiting this link.
Determine the Asymptote of the Horizontal from a Graph
One thing is consistent across all these charts. An asymptote on the horizontal plane is a line that the curve approaches in a horizontal direction. A HA, however, must never make direct contact with the curve itself (though it is permitted to traverse the curve).
If f(x) = 2x / (x - 3), then determine its horizontal asymptote.
The formula for minimizing f(x) is: lim x f(x) = lim x 2x / (x - 3) = lim x 2x / [x (1 - 3/x)] = lim x 2 / (1 - 3/x) = 2 / (1 - 0).
That means the HA of the function is y = 2. This leads us to the next step: determining the other boundary.
Limit of f(x) = 2x / (x - 3) = 2x / [x (1 - 3/x)] = 2 / (1 - 3/x) = 2 / (1 0)
We have a 2 once more, so we can rewrite the HA from before as y = 2.
That means y = 2 is the only possible horizontal asymptote for the function. Observe the visual evidence here.
In fact, the range of a rational function can be determined by utilizing the horizontal asymptote. To prove that the HA is not included in the function's graph, we merely exploit this fact. Given the above graph, we can say that f(x) has a range of [yR|y2].
Should we always use the limits to identify the HA? Couldn't there be a simple solution to this? Some shortcuts and techniques for locating the horizontal asymptotes of various types of functions are provided below.
Locating the Asymptote on the Horizon for a Rational Function
Only one horizontal asymptote can exist in a rational function. While we can use limits to locate the HAs, there are some shortcuts we can take to make locating the HAs of rational functions much easier.
- No HA exists in the function if the numerator degree is greater than the denominator degree.
- One HA (y = 0) is present in the function if the numerator degree is greater than the denominator degree.
- One HA (y = the ratio of the leading coefficients of numerator and denominator) is present in the function if the numerator degree is equal to the denominator degree.
The degree of the numerator equals the degree of the denominator (= 1) in the previous section's example (where f(x) = 2x / (x - 3)). Therefore, the HA of f(x) is y = 21 = 2. In spite of applying the limits, we still obtained the same result. The following are a few more instances of this:
- There is no HA in y = (x2 3) / (2x).
- Y = (2x) / (x2 + x3) is the identity (HA), so y = 0.
- If you divide y by two and two, you get x three, so the HA of y = (2x2) / (x2 3) is 2.
Exponential Function: Locating the Horizontal Asymptote
There is always only one horizontal asymptote for an exponential function. The original form of the exponential function, f(x) = bx, can be transformed into other forms, such as f(x) = abkx c. Asymptote on the horizontal plane where the parent exponential function is transposed vertically is denoted by the symbol 'c'. As a result:
- When f(x) = bx, the HA is y = 0.
- An HA for f(x) = abkx c is y = c.
In light of the preceding information, the horizontal asymptote of the exponential function f(x) = 4x 2 is y = 2 (or, more precisely, y = lim x - 4x 2 = 0 2 = 2).
Instances are provided below.
- F(x) = -2x -3 has an HA of y = -3.
- F(x) = 3-x5 is the HA of f(y) = 5
- F(x) = 0 is the f(x) homotopy asymptote. 52x - (2/3) is y = -2/3
Relating to Asymptotes on the Horizontal
Summarize all the rules we have seen for horizontal asymptotes so far.
- In order to locate the horizontal asymptote of a rational function, one must first determine the numerator and denominator degrees. It can be shown that HA is y = 0 if and only if n is smaller than d. No HA exists if (n > d).
A HA is a ratio of the leading coefficients, denoted by y = ratio of leading coefficients if n = d.
- An exponential function f(x) = abkx c has a horizontal asymptote at y = c.
- Since the limits of polynomial functions as x tends to or - do not give real numbers, there can be no horizontal asymptote for a polynomial function (such as f(x) = x 3 or f(x) = x2-2x 3).
- To determine the horizontal asymptote for any function other than those listed above, we use the standard method of imposing limits as x and x -.
Considerations on the Horizontal Asymptote
Determine the horizontal asymptote of y = (3x2 2x)/(x 1) as an example 1.
The numerator degree here is 2, for example.
One is the lowest possible denominator degree.
Hence, the numerator's degree is greater than the denominator's.
It follows that f(x) is not HA.
The given function does not contain any HA.
Example 2: If the HA of f(x) = 2x - k is y = 3, then what is the value of k? Use the rules for horizontal asymptotes to find the answer.
The HA of an exponential function is known to be the result of inverting the function.
That's why we can write y = -k as the HA of f(x) = 2x - k.
However, it is known that y = 3 is the HA of f(x).
Therefore, k = -3 or -k = 3
Solved: k = -3
Determine the HAs of the function f(x) = (frac x 1sqrt x2-1) in Example 3.
No classification can be made for the given function. So the rules for locating HA based on horizontal asymptotes do not apply here. Therefore, we employ limits to determine HA.
For any value of x, we can write lim x f(x) = lim x (fracx 1sqrtx2-1).
= lim x (frac x left(1 frac x right)|x| sqrt 1-frac x2)
Since x, |x| = x in this context. Thus, the preceding procedure is
= lim x (frac xleft(1 frac xright)x sqrt1-frac1 x2) = lim x (frac xleft(1 frac xright)x sqrt1-frac1 x2 = 1 / (1 - 0)
Therefore, the HA of the function is at y = 1. That other bound is what we're going to find now.
f(x) lim x - f(x) = lim x - (frac x 1 sqrt x - 1)
= lim x - (frac x left(1 frac x right) sqrt x - (frac x squared)/(frac x cubed)
In this case, x-, so |x| = -x Therefore, the preceding procedure is
= lim x (frac left(1 frac1xright)-sqrt1-frac1x2) = 1 / (-(1 - 0))
So the other HA is y = -1.
The asymptotes of the function on the horizontal axis are y = 1 and y = -1.
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A Look at Some Frequently Asked Questions About the Horizontal Asymptote
An example of a function with a horizontal asymptote (HA) is y = f(x), where f(x) is the limit of y as x approaches zero or x approaches -x. An HA count of two is the maximum for a single function.
Just what are these Horizontal Asymptote Rules that everyone keeps talking about?
The formulas y = lim x f(x) and/or y = lim x - f(x) are commonly used to provide a general rule for determining the HA of y = f(x). However, the HA of certain functions can be located with the following tips in mind:
- To determine the HA of a rational function
when the degree of the numerator is smaller than the degree of the denominator, y = 0.
has no meaning when the degree of the numerator is higher than that of the denominator
When the degrees are the same, does k equal the ratio of the leading coefficients, and y equals k?
- For the exponential function f(x) = ax k, the HA is y = k.
When comparing vertical and horizontal asymptotes, what are the key differences?
In mathematics, asymptotes are lines that a function "appears" to be on, but which it is not. In mathematics, a horizontal asymptote is a straight line with the equation y = k. The formula for a vertical asymptote is x = k, which describes a straight line.
An Explanation of the Method for Finding the Horizontal Asymptote
We can use the formulas y = lim x f(x) and y = lim x - f(x) to determine the horizontal asymptotes of the function y = f(x). Throw out the result of any of these bounds if it is not a real number.
Identifying the Asymptote at the Horizon for a Rational Function
Finding the HA of a rational function y = f(x) is greatly aided by the numerator and denominator degrees.
- The HA formula is y = ratio of leading coefficients if and only if n = d.
- When n is larger than d, f(x) is not HA if and only if
- The HA is y = 0 if and only if n is smaller than d.
Determine the Exponential Function's Horizontal Asymptote
The standard form of an exponential function is y = ax. The function takes the form y = ax + k if there is a vertical transformation. The HA for this system is y = k.
A Simple Method for Locating the Bounds of a Function Using Its Horizontal Asymptote
In the special case of a rational function, the range can be found by locating the horizontal asymptote. The HA of the function f(x) = (2x) / (x2)+1 is y = 0, and its range is [y R | y 0].
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