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

Problem

ƒ(x)=(4*x)/(x2−4)

Solution

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

x2−4=0

(x−2)*(x+2)=0

x=2,x=−2

2. Verify the numerator at these values to ensure they are not holes.

4*(2)=8≠0

4*(−2)=−8≠0

Vertical Asymptotes: *x=2,x=−2

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

Degree of numerator *(4*x)=1

Degree of denominator *(x2−4)=2

4. Apply the rule for horizontal asymptotes where the degree of the denominator is greater than the degree of the numerator.

Horizontal Asymptote: *y=0

5. Check for slant asymptotes by noting that they only occur when the degree of the numerator is exactly one higher than the degree of the denominator.

1<2

Slant Asymptote: None

Final Answer

Vertical Asymptotes: *x=2,x=−2; Horizontal Asymptote: *y=0

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