Find the Asymptotes f(x)=(x^2-2x-8)/(x^2-9)

Problem

ƒ(x)=(x^2−2*x−8)/(x^2−9)

Solution

1. Identify vertical asymptotes by finding the values of x that make the denominator zero while the numerator remains non-zero.

x^2−9=0

(x−3)*(x+3)=0

x=3,x=−3

2. Check for holes by evaluating the numerator at these x values.

3^2−2*(3)−8=−5

(−3)^2−2*(−3)−8=7

Since neither value makes the numerator zero, both are vertical asymptotes.

3. Identify horizontal asymptotes by comparing the degrees of the numerator and the denominator.

degree of numerator=2

degree of denominator=2

4. Apply the rule for equal degrees, where the horizontal asymptote is the ratio of the leading coefficients.

y=1/1

y=1

5. Determine slant asymptotes by checking if the degree of the numerator is exactly one higher than the denominator.
   Since the degrees are equal, there is no slant asymptote.

Final Answer

Vertical Asymptotes: *x=3,x=−3; Horizontal Asymptote: *y=1

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