Find the Asymptotes f(x)=(x^2+x-6)/(x^2+3x-10)

Problem

ƒ(x)=(x2+x−6)/(x2+3*x−10)

Solution

1. Factor the numerator and the denominator to identify potential vertical asymptotes and holes.

ƒ(x)=((x+3)*(x−2))/((x+5)*(x−2))

2. Identify holes by finding common factors in the numerator and denominator.

Since *(x−2)* is a common factor, there is a hole at *x=2.

3. Simplify the function by canceling the common factor to find the reduced form.

ƒ(x)=(x+3)/(x+5)* for *x≠2

4. Find vertical asymptotes by setting the denominator of the simplified function to zero.

x+5=0

x=−5

5. Find horizontal asymptotes by comparing the degrees of the numerator and denominator in the original function.

Since the degrees are equal, *y=leading coefficient of numerator/leading coefficient of denominator

y=1/1

y=1

Final Answer

Vertical Asymptote: *x=−5, Horizontal Asymptote: *y=1

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