Question

What is the horizontal asymptote of a function if m > n?

a) y = 1
b) y = 0
c) There is no horizontal asymptote
d) x = 0
e) Other

Answer 1

The horizontal asymptote of a function when the degree of the numerator (m) is greater than the degree of the denominator (n) is that there is no horizontal asymptote (option c). As m becomes larger than n, the values of the function increase or decrease without approaching a finite limit.

Step-by-step explanation:

The question is asking about the behavior of a function as it extends towards infinity, particularly what its horizontal asymptote would be. In mathematical terms, a horizontal asymptote is a y-value that the function approaches but never quite reaches as the x-values head towards infinity (positively or negatively).

When considering the degrees of the numerator (m) and the denominator (n) in a rational function (a fraction where both the numerator and the denominator are polynomials), the relationship between m and n can determine the horizontal asymptote:

• If m is less than n, the horizontal asymptote is y = 0.
• If m is equal to n, the horizontal asymptote is found by dividing the leading coefficients.
• If m is greater than n, there is no horizontal asymptote.

Hence, if m > n, the correct answer would be c) There is no horizontal asymptote.