Given the following rational functions,find:a.)the horizontal asymptote(s), b.)the vertical asymptote(s), if any,and c.) the oblique asymptote(),if any. f(x)=x(x-17)^2/(x+12)^3

f(x) = x(x-17)^2 / (x+12)^3

a)
Horizontal as:
consider what happens when x becomes large
When expanded both numerators and denominators lead with +x^3
so as x ---> ∞ f(x) = x^3 .../x^3 ... = 1

e.g.
let x = 100,000 , f(100,000) = .9993..
let x = -100,000 , f(-100,000) = 1.0007...

so the H.A. is y = 1

b)
vertical asymptotes are caused by the denominator becoming zero
x+12=0
x = -12 is your V.A.

c) no oblique

Want to let you know that the H.A. is WRONG by the way

To find the horizontal asymptote(s) of a rational function, we need to look at the degrees of the numerator and denominator polynomials. Let's evaluate each part separately.

a.) Horizontal Asymptote(s):
1. Start by comparing the degrees of the numerator and denominator polynomials.
- The numerator polynomial is of degree 3 (x(x-17)^2).
- The denominator polynomial is of degree 3 ((x+12)^3).

2. Since the degrees are the same, the horizontal asymptote can be determined by dividing the leading coefficients of the numerator and denominator polynomials.
- The leading coefficient of the numerator is 1.
- The leading coefficient of the denominator is 1.

So, the equation of the horizontal asymptote is y = 1.

b.) Vertical Asymptote(s):
To find the vertical asymptotes of a rational function, we look for any values of x that make the denominator equal to zero.

1. Set the denominator equal to zero and solve for x.
(x + 12)^3 = 0

2. In this case, the denominator (x + 12) raised to the power of 3 gives a repeated root of x = -12.
Thus, we have a vertical asymptote at x = -12.

c.) Oblique Asymptote:
To determine the oblique asymptote, we need to check if the degrees of the numerator and denominator polynomials differ by exactly one.

1. The degree of the numerator polynomial is 3.
2. The degree of the denominator polynomial is 3.

Since the degrees are the same, there is no oblique asymptote in this case.

In summary:
a.) The horizontal asymptote is y = 1.
b.) The vertical asymptote is x = -12.
c.) There is no oblique asymptote for the given rational function.