Questions: Find the horizontal vertical asymptotes of f(x). f(x) = 4x / (x+4)

Solution Steps

To find the horizontal and vertical asymptotes of the function \( f(x) = \frac{4x}{x+4} \):

1. Vertical Asymptote: Set the denominator equal to zero and solve for \( x \).
2. Horizontal Asymptote: Compare the degrees of the numerator and the denominator. If they are the same, the horizontal asymptote is the ratio of the leading coefficients.

To find the vertical asymptote, we set the denominator of the function equal to zero and solve for \( x \): \[ x + 4 = 0 \implies x = -4 \] Thus, the vertical asymptote is \( x = -4 \).

To find the horizontal asymptote, we compare the degrees of the numerator and the denominator. Both the numerator and the denominator are linear (degree 1). Therefore, the horizontal asymptote is the ratio of the leading coefficients: \[ \frac{4}{1} = 4 \] Thus, the horizontal asymptote is \( y = 4 \).

Final Answer