Find the Asymptotes f(x)=x/(x^2-1)

Problem

ƒ(x)=x/(x2−1)

Solution

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

x2−1=0

(x−1)*(x+1)=0

x=1,x=−1

2. Verify vertical asymptotes by checking if the numerator x is non-zero at these points. Since 1≠0 and −1≠0 both are vertical asymptotes.

x=1

x=−1

3. Identify horizontal asymptotes by comparing the degrees of the numerator and the denominator. The degree of the numerator x is 1 and the degree of the denominator x2−1 is 2

degree of numerator<degree of denominator

4. Apply the rule for horizontal asymptotes where the degree of the denominator is greater than the degree of the numerator. This implies the limit as x approaches infinity is 0

(lim_x→∞)(x/(x2−1))=0

y=0

Final Answer

Vertical: *x=1,x=−1; Horizontal: *y=0

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