Find the Asymptotes f(x)=(x^2-4x)/(x^2-4)

Problem

ƒ(x)=(x2−4*x)/(x2−4)

Solution

1. Factor the numerator and the denominator to identify common factors and points of interest.

ƒ(x)=(x*(x−4))/((x−2)*(x+2))

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

(x−2)*(x+2)=0

x=2,x=−2

3. Check for holes by seeing if any factors cancel out. Since no factors are common to both the numerator and denominator, there are no holes.

Vertical Asymptotes: *x=2,x=−2

4. Identify horizontal asymptotes by comparing the degrees of the numerator and the denominator. Since both are degree 2, the asymptote is the ratio of the leading coefficients.

(lim_x→∞)((x2−4*x)/(x2−4))=1/1=1

Horizontal Asymptote: *y=1

5. Check for slant asymptotes by comparing degrees. Since the degree of the numerator is not exactly one greater than the degree of the denominator, there is no slant asymptote.

Final Answer

Vertical Asymptotes: *x=2,x=−2; Horizontal Asymptote: *y=1

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