Horizontal Asymptotes and Polynomial Degrees

Nancy explains how to find horizontal asymptotes for rational expressions by considering three cases based on the degrees of the polynomials involved. She describes each case: when the degree on top is less than, equal to, or greater than the degree on the bottom. For each scenario, she provides the rule for determining the horizontal asymptote. Nancy emphasizes that understanding these rules is sufficient and that formal limit definitions are unnecessary. She concludes by encouraging viewers to remember these simple rules.

What is the degree of the polynomial x^2 + 3x + 4?

• 1

• 2

• 3

• 4

If the degree of the numerator is less than the degree of the denominator, what is the horizontal asymptote?

• None

• y = x

• y = 1

• y = 0

In the first case, what is the horizontal asymptote when the degree of the numerator is 1 and the degree of the denominator is 2?

• y = 1

• y = 0

• y = 2

• None

When the degrees of the numerator and denominator are equal, how do you find the horizontal asymptote?

• By multiplying the coefficients

• By dividing the leading coefficients

• By adding the coefficients

• By subtracting the coefficients

What is the horizontal asymptote if the degrees of the numerator and denominator are both 2, and the leading coefficients are 1 and 3 respectively?

• None

• y = 1

• y = 3

• y = 1/3

What happens to the horizontal asymptote when the degree of the numerator is greater than the degree of the denominator?

• y = x

• None

• y = 1

• y = 0

In the third case, what type of asymptote might exist if the degree of the numerator is greater than the degree of the denominator?

• Horizontal

• Vertical

• Oblique

• None

What is the key takeaway regarding the use of limits in finding horizontal asymptotes?

• Use limits for all asymptotes

• Use limits only for vertical asymptotes

• Never use limits

• Always use limits

Which of the following is NOT a case for determining horizontal asymptotes?

• Degree of numerator = degree of denominator

• Degree of numerator < degree of denominator

• Degree of numerator > degree of denominator

• Degree of numerator = 0

What should you remember to determine the horizontal asymptote of a rational function?

• The limit definition

• The three cases

• The polynomial coefficients

• The x-intercepts