Find the Asymptotes f(x)=(x^2+x-30)/(x-7)

Problem

ƒ(x)=(x2+x−30)/(x−7)

Solution

1. Identify the vertical asymptote by finding the values of x that make the denominator zero while the numerator is non-zero.

x−7=0

x=7

2. Check for horizontal asymptotes by comparing the degrees of the numerator and denominator. Since the degree of the numerator (2 is exactly one greater than the degree of the denominator (1, there is no horizontal asymptote, but there is a slant (oblique) asymptote.

3. Perform polynomial long division to find the slant asymptote by dividing x2+x−30 by x−7

(x2+x−30)/(x−7)=x+8+26/(x−7)

4. Determine the slant asymptote from the quotient of the division, ignoring the remainder as x approaches infinity.

y=x+8

Final Answer

Vertical Asymptote: *x=7, Slant Asymptote: *y=x+8

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