Find the Asymptotes f(x)=(x-4)/(x(x-4))

Problem

ƒ(x)=(x−4)/(x*(x−4))

Solution

1. Identify the domain by finding values that make the denominator zero. Setting x*(x−4)=0 gives x=0 and x=4

2. Simplify the function by canceling common factors in the numerator and denominator. For x≠4 the expression reduces to ƒ(x)=1/x

3. Determine vertical asymptotes by checking where the simplified function is undefined. Since the factor (x−4) canceled out, x=4 is a removable discontinuity (a hole). The remaining factor in the denominator is x so the vertical asymptote is x=0

4. Determine horizontal asymptotes by comparing the degrees of the numerator and denominator. Since the degree of the numerator (1) is less than the degree of the denominator (2), the horizontal asymptote is y=0

Final Answer

Vertical Asymptote: *x=0, Horizontal Asymptote: *y=0

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