what is the horizontal asymptote for y=(x-2)/(x^2-9)?
To find the horizontal asymptote for the function y = (x-2)/(x^2-9), we need to examine the behavior of the function as x approaches positive or negative infinity.
Step 1: Determine the highest power of x in the numerator and denominator. In this case, the highest power of x is x^2 in the denominator.
Step 2: Divide all terms in the function by x^2. Doing this will help us determine the limit as x approaches infinity.
y = (x-2)/(x^2-9) ⇒ y = (x/x^2 - 2/x^2)/(x^2/x^2 - 9/x^2)
Simplifying, we get: y = (1/x - 2/x^2)/(1 - 9/x^2)
Step 3: Take the limit as x approaches infinity.
lim[x→∞] (1/x - 2/x^2)/(1 - 9/x^2) = (0 - 0)/(1 - 0) = 0/1 = 0
Step 4: Interpret the limit.
As x approaches positive or negative infinity, the function y approaches 0. Therefore, the horizontal asymptote for y = (x-2)/(x^2-9) is y = 0.