Find the Asymptotes f(x)=(x^2-100)/((x-9)(x+3))

Problem

ƒ(x)=(x2−100)/((x−9)*(x+3))

Solution

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

(x−9)*(x+3)=0

x=9,x=−3

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

9−100=−19≠0

(−3)2−100=−91≠0

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

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

ƒ(x)=(x2−100)/(x2−6*x−27)

4. Apply the rule for horizontal asymptotes when the degrees are equal. The asymptote is the ratio of the leading coefficients.

y=1/1=1

Final Answer

Vertical Asymptotes: *x=9,x=−3; Horizontal Asymptote: *y=1

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