Math: Determining values of a & b that will allow for certain asymptotes?

Question

How can I find the values of a & b such that (a*x^2 - 3*x + 12) / (2*x * (x - b)) has the vertical asymptote x = 9 and the horizontal asymptote y = 7?
I have the answers, but I need to know how to calculate them. Thank you!

But the value of a should be equal to 14...?

Answer 1

Normally, to find vertical asymptotes, you equate the denominator to zero and then solve for x. In this case, we're given the asymptote x = 9, so we have to make

2(9)(9 - b) = 0

All we have to do is solve for b.

18(9-b) = 0
162 - 18b = 0
162 = 18b
Therefore, b = 9

To determine the horizontal asymptote, we have to solve for the limit as x approaches infinity, and we want this to equal 7.

Thus, we have to solve

lim ( [(a*x^2 - 3*x + 12) / (2*x * (x - b))]
x -> infinity

Let's first pull out that pesky constant in the denominator. When we pull it out of the limit, it becomes 1/2

(1/2)* lim ( [(a*x^2 - 3*x + 12) / (x^2 - xb))]
x -> infinity

Normally when solving fractions and limits to infinity, we divide by the highest power of x. In this case, it's x^2. If we put everything on the top and bottom over x^2, some x terms will cancel out and some will be left behind.

(1/2) * lim ( [(a - 3/x + 12/x^2) / (1 - b/x) ] )
x -> infinity

Now, we can individually solve each term as x approaches infinity

(1/2) [ (a - 0 + 0) / (1 - 0) ] = (1/2)(a/1) = a/2

However, we know what the limit SHOULD be, because the limit should be the horizontal asymptote 9. Therefore, we just equate a/2 to 9

a/2 = 9
Therefore, a = 18

So our two values for a and b are 18 and 9 respectively

Answer 2

if a=8 and b= -2 2a + 3b replace a with 8 and b with -2 and placed the two into ( ) to coach that it going to be more desirable 2(8) +3(-2) 2(8)=sixteen and 3(-2)=-6 now sixteen+ -6 with equivalent out as 10 in this equation sixteen+-6 may even get replaced to sixteen-6=10 the respond is 10! =]