Problem 1 Problem 2 Find the horizontal asymptote if it exis

Problem 1

Problem #2

Find the horizontal asymptote, if it exists, of the rational function below. If there is no horizontal asymptote, enter NONE.

Solution

1) a) Lt x-- inf (6x(x^2+9)/(-9 -9x^4)

= 6x*x^2( 1+9/x^2)/x^4(-9 -9/x^4)

= 6(1+9/x^2)/x( -9 -9/x^4)

plug x= inf

= 6/inf = 0

b) x --- -inf (4x(x+3)(x-9)/(7- 2x^3)

= 4x^2(1+3/x)x(1-9/x))(x^3( 7/x^3-2)

= 4x^3(1+3/x)(1-9/x)/x^3(7/x^3 -2)

= 4(1+3/x)(1-9/x)/(7/x^3 -2)

plug x= -inf

= 4(1+0)(1-0)/(0-2)

= -2

2) f(x) = 5 -10/x^2 -2x/(x-8)

= {5(x-8)x^2 -10(x-8) -2x^3}/x^2(x-8)

= { 3x^3 -8x^2 -10x +80}/x^2(x-8)

When the degrees of the numerator and the denominator are the same, then the horizontal asymptote is found by dividing the leading terms, so the asymptote is given by:

y = (numerator\'s leading coefficient) / (denominator\'s leading coefficient)

y =3x^3/x^3 = 3

y = 3