Find the Asymptotes f(x)=(2x^2+1)/(x^2-4)

Problem

ƒ(x)=(2*x2+1)/(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. Identify horizontal asymptotes by comparing the degrees of the numerator and the denominator. Since both are degree 2, the asymptote is the ratio of the leading coefficients.

y=2/1

y=2

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

Final Answer

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

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