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

Problem

ƒ(x)=(2*x2)/(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 horizontal asymptotes by comparing the degrees of the numerator and the denominator. Since both are degree 2, calculate the ratio of the leading coefficients.

(lim_x→∞)((2*x2)/(x2−1))=2

y=2

3. Check for oblique asymptotes by comparing degrees. Since the degree of the numerator is not exactly one greater than the degree of the denominator, there are no oblique asymptotes.

Final Answer

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

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