Questions: Find the horizontal and vertical asymptotes of f(x). Use limits to describe the corresponding behavior. f(x) = (7x+4)/(x^2-x) Select the correct choice for the vertical asymptote below and, if necessary, fill in any answer box that completes your choice. A. The function has one vertical asymptote. The equation for the asymptote is (Type an equation.) B. The function has two vertical asymptotes. The equation for the leftmost asymptote is (Type equations.) and the equation for the rightmost asymptote is I. C. The function has three vertical asymptotes. The equation for the leftmost asymptote is , the equation for the middle asymptote is . and the equation (Type equations.) D. There is no vertical asymptote.

Solution Steps

To find the vertical asymptotes of the function \( f(x) = \frac{7x+4}{x^2-x} \), we need to determine where the denominator is zero, as these are the points where the function is undefined and may have vertical asymptotes. For horizontal asymptotes, we compare the degrees of the numerator and the denominator. Since the degree of the denominator is higher than the numerator, the horizontal asymptote is at \( y = 0 \).

To find the vertical asymptotes of the function \( f(x) = \frac{7x + 4}{x^2 - x} \), we set the denominator equal to zero:

\[ x^2 - x = 0 \]

Factoring gives:

\[ x(x - 1) = 0 \]

Thus, the solutions are:

\[ x = 0 \quad \text{and} \quad x = 1 \]

This indicates that there are two vertical asymptotes at \( x = 0 \) and \( x = 1 \).

To determine the horizontal asymptote, we compare the degrees of the numerator and the denominator. The degree of the numerator \( 7x + 4 \) is 1, and the degree of the denominator \( x^2 - x \) is 2. Since the degree of the denominator is greater than that of the numerator, the horizontal asymptote is:

\[ y = 0 \]

Final Answer