Finding Horizontal Asymptotes

Description/Explanation/Highlights

Video Description

In this video we will learn how to find horizontal asymptotes in a rational function.

Steps and Key Points to Remember

To find a horizontal asymptote in a rational function, follow these steps:

  1. Horizontal asymptotes occur when a graph approaches a horizontal line but never reaches it.
  2. To find a horizontal asymptote, we must first find the degrees of both the numerator and denominator and compare them.
  3. To find the degree of the numerator or denominator, use the largest exponent on a variable.
  4. Compare the degree of the numerator to the degree of the denominator.
  5. If the degree of the numerator is greater, there is no horizontal asymptote.
  6. If the degree of numerator is less than the degree of the denominator, the horizontal asymptote is at \(y=0\).
  7. If the degree of the numerator and denominator are the same, divide the leading coefficient of the numerator by the leading coefficient of the denominator to get the horizontal asymptote. Note: Be sure the polynomials are in order from largest exponent to smallest exponent before finding the leading coefficient.
  8. In the example \(\frac{\textstyle 3x^2+9}{\textstyle 4x^3+2x}\), the largest exponent in the numerator is 2, so the degree of the numerator is 2. The degree of the denominator is 3 since the largest exponent is 3. Since the degree of the numerator is less than the degree of the denominator the horizontal asymptote is at \(y=0\).
  9. In the example, \(\frac{\textstyle 4x^3-2x+1}{\textstyle 2x^2-1}\), the degree of the numerator is 3 and the degree of the denominator is 2. Since the degree of the numerator is greater, there is no horizontal asymptote.
  10. In the example, \(\frac{\textstyle 6x^2+2x-1}{\textstyle 2x^2+4}\), the degrees of both the numerator and denominator equal 2. Since the degrees are the same, divide the leading coefficient of the numerator by the leading coefficient of the denominator. Since the polynomials are in order, divide 6 by 2. The horizontal asymptote is at \(y=3\).

Here are some key points to keep in mind when finding horizontal asymptotes for rational equations.

  • Horizontal asymptotes are found by comparing the degree of the numerator to the degree of the denominator.
  • If the degree of the numerator is greater, there is no horizontal asymptote.
  • If the degree of the denominator is greater, the horizontal asymptote is at \(y=0\).
  • If the degrees are the same, divide the lead coefficient of the numerator by the lead coefficient of the denominator and set y equal to the answer to find the location of the the horizontal asymptote.
  • Be sure the polynomials in the numerator and denominator are in descending exponential order before finding the leading coefficient.
  • Horizontal asymptotes always are expressed as a “\(y=\)” line and are usually graphed as a dotted reference line on the graph. 

Video Highlights

  • 00:00 Introduction
  • 00:15 Definition and explanation of horizontal asymptotes
  • 00:35 \(y=\frac{\textstyle 3x^2+9}{\textstyle 4x^3+2x}\) example of finding and comparing degrees
  • 01:26 \(y=\frac{\textstyle 4x^3-2x+1}{\textstyle 2x^2-1}\) example resulting in no horizontal asymptote
  • 02:50 \(y=\frac{\textstyle 3x+2}{\textstyle x^2-3x+1}\) example when the horizontal asymptote is \(y=0\)
  • 03:45 \(y=\frac{\textstyle 6x^2+2x-1}{\textstyle 2x^2+4}\) example of finding a horizontal asymptote when the degrees of the numerator and denominators are the same.
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