Question

1. horizontal versus oblique asymptote

a. under what circumstances does a rational function have a horizontal asymptote of y=0

b. under what circumstances does a rational function have a horizontal asymptote that is not y=0

c. under what circumstances does a rational function have an oblique asymptote

d. find horizontal and /or oblique asymptote for each of the following

i. f(x)=x^2-3x+8/x+1

ii. g(x)=4/x+1

iii. h(x)=3x+4/x+1

e. what are the vertical asymptote for each of the functions f, g, and h in problem 1d?

Answer 1

1.a. When a rational function f(x) = p(x)/q(x),where p(x) and q(x) are polynomials in a single variable x and when n < m, where n and m are the degrees of the numerator and the denominator respectively, then the line y = 0, i.e. the X-Axis is a horizontal asymptote.

b. when n = m or, n > m, the rational function f(x) = p(x)/q(x) does not have the line y = 0 as a horizontal asymptote.

c. If n=m+1, there is an oblique or slant asymptote.

d. i. Here f(x) = (x²-3x+8)/(x+1). The degree of the numerator is higher than the degree of the denominator by 1 so there will be an oblique or slant asymptote. Further, (x²-3x+8) = x(x+1)-4(x+1) +12 = (x+1)(x-4)+12 so that f(x) = (x-4)+ 12/(x+1). Hence, the line y = x-4 is an oblique or slant asymptote.

ii. Here, g(x) = 4/(x+1). Since the degree of the numerator is lower than the degree of the denominator, hence the line y = 0 is a horizontal asymptote.

iii. Here, h(x)=(3x+4)/(x+1). Since the degrees of the numerator and the denominator are equal, there is no horizontal or oblique asymptote.

e. When a rational function f(x) = p(x)/q(x),where p(x) and q(x) are polynomials in a single variable x, the equations of the vertical asymptotes can be found by finding the roots of q(x). Thus, x = -1 is a vertical asymptote for each of f(x),g(x) and h(x).