Find the Asymptotes f(x)=(x^2-9)/(x-3)

Problem

ƒ(x)=(x2−9)/(x−3)

Solution

1. Factor the numerator of the function using the difference of squares formula.

x2−9=(x−3)*(x+3)

2. Simplify the expression by canceling the common factor in the numerator and denominator.

ƒ(x)=((x−3)*(x+3))/(x−3)

ƒ(x)=x+3,x≠3

3. Identify vertical asymptotes by checking for values where the denominator is zero after simplification. Since the factor (x−3) cancels out, there is a removable discontinuity (a hole) at x=3 rather than a vertical asymptote.

Vertical Asymptotes: None

4. Identify 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.

Horizontal Asymptotes: None

5. Identify slant (oblique) asymptotes. A slant asymptote exists if the simplified form is a linear equation. However, because the function simplifies exactly to a linear polynomial y=x+3 with a hole, the graph is a line, not a curve approaching a line.

Slant Asymptotes: None

Final Answer

Asymptotes: None

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