Question

Identify the horizontal and vertical asymptotes, if any, of the given function. x² – 4 f(x) = 4.x - 4 Separate multiple answers by commas. Enter DNE if an asymptote does not exist. y a) Horizontal asymptote(s): b) Vertical asymptote(s): =

Answer 1

Explanation:

a) Horizontal asymptote(s): There is a horizontal asymptote when the degree of the numerator is equal to the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 1. Therefore, there is no horizontal asymptote.

b) Vertical asymptote(s): There is a vertical asymptote when the denominator is equal to zero, and the numerator is not equal to zero at the same point. In this case, the denominator is x - 2, which equals zero when x = 2. Therefore, there is a vertical asymptote at x = 2.

Note that the function can be written as:

f(x) = (4x - 4) / (x - 2)

This form makes it clear that there is a vertical asymptote at x = 2 and no horizontal asymptote.