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

Problem

ƒ(x)=(x2+1)/(x2−1)

Solution

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

x2−1=0

(x−1)*(x+1)=0

x=1,x=−1

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

(lim_x→∞)((x2+1)/(x2−1))=1/1

y=1

3. Check for oblique asymptotes by noting that the degree of the numerator is not exactly one higher than the degree of the denominator. Since the degrees are equal, there is no oblique asymptote.

Final Answer

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

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