Find the Asymptotes f(x)=(x^2-16)/(x-4)

Problem

ƒ(x)=(x2−16)/(x−4)

Solution

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

x2−16=(x−4)*(x+4)

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

ƒ(x)=((x−4)*(x+4))/(x−4)

ƒ(x)=x+4,x≠4

3. Identify vertical asymptotes by checking for values where the denominator is zero after simplification. Since the factor (x−4) canceled out, there is a removable discontinuity (a hole) at x=4 but no vertical asymptote.

4. Identify horizontal or slant asymptotes by looking at the simplified form. The simplified function ƒ(x)=x+4 is a linear function, which does not approach a constant value as x→∞ or x→−∞ Therefore, there are no horizontal asymptotes.

5. Determine if there is a slant (oblique) asymptote. A slant asymptote occurs when the degree of the numerator is exactly one higher than the degree of the denominator. However, because the division results in a polynomial with no remainder, the graph is a line with a hole, not a curve approaching a line.

Final Answer

No asymptotes

Want more problems? Check here!