Find the Asymptotes f(x)=(x^2+x-12)/(x^2+2x-15)

Problem

ƒ(x)=(x2+x−12)/(x2+2*x−15)

Solution

1. Factor the numerator and the denominator to identify common factors and potential discontinuities.

ƒ(x)=((x+4)*(x−3))/((x+5)*(x−3))

2. Identify the hole by finding the value of x where both the numerator and denominator are zero.

x−3=0⇒x=3

3. Simplify the function by canceling the common factor, noting that the simplified form is valid for x≠3

ƒ(x)=(x+4)/(x+5)

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

x+5=0⇒x=−5

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

(lim_x→∞)((x2+x−12)/(x2+2*x−15))=1

y=1

Final Answer

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

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