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

Problem

ƒ(x)=(2*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 vertical asymptotes by checking if the numerator 2*x is non-zero at these points.

2*(2)=4≠0

2*(−2)=−4≠0

Vertical asymptotes: *x=2,x=−2

3. Identify horizontal asymptotes by comparing the degree of the numerator to the degree of the denominator.

Degree of numerator *(2*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.

(lim_x→∞)((2*x)/(x2−4))=0

Horizontal asymptote: *y=0

Final Answer

Asymptotes: *x=2,x=−2,y=0

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