Find the Asymptotes f(x)=(x^2-4)/(x^2-3x+2)

Problem

ƒ(x)=(x^2−4)/(x^2−3*x+2)

Solution

1. Factor the numerator and the denominator to identify potential discontinuities and simplify the function.

ƒ(x)=((x−2)*(x+2))/((x−2)*(x−1))

2. Simplify the expression by canceling the common factor (x−2) noting that a hole exists at x=2

ƒ(x)=(x+2)/(x−1)

3. Find the vertical asymptote by setting the remaining denominator equal to zero.

x−1=0

x=1

4. Find the horizontal asymptote by comparing the degrees of the numerator and denominator or by taking the limit as x approaches infinity.

(lim_x→∞)((x^2−4)/(x^2−3*x+2))=1

y=1

Final Answer

Vertical Asymptote: *x=1, Horizontal Asymptote: *y=1

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