Write an fquation that defines a rational function with a vertical asymptote at $x=1$ and a hocizontat asymptote at $y=4$ Choose the correct answer below. A. $y=\frac{x-4}{x-1}$ B. $y=\frac{4 x-5}{x-1}$ c. $y=\frac{5 x-4}{x-1}$ D. $y=\frac{x-5}{x-4}$

Step 1 ：A rational function has a vertical asymptote at \(x=a\) if the denominator of the function is zero at \(x=a\). Similarly, a rational function has a horizontal asymptote at \(y=b\) if the ratio of the leading coefficients of the numerator and the denominator is \(b\).

Step 2 ：In this case, we are looking for a function with a vertical asymptote at \(x=1\) and a horizontal asymptote at \(y=4\). This means that the denominator of the function should be zero at \(x=1\) and the ratio of the leading coefficients of the numerator and the denominator should be 4.

Step 3 ：Let's check each option one by one.

Step 4 ：For option A, \(y=(x - 4)/(x - 1)\), the limit as \(x\) approaches 1 is \(-\infty\) and the limit as \(x\) approaches \(\infty\) is 1.

Step 5 ：For option B, \(y=(4*x - 5)/(x - 1)\), the limit as \(x\) approaches 1 is \(-\infty\) and the limit as \(x\) approaches \(\infty\) is 4.

Step 6 ：For option C, \(y=(5*x - 4)/(x - 1)\), the limit as \(x\) approaches 1 is \(\infty\) and the limit as \(x\) approaches \(\infty\) is 5.

Step 7 ：For option D, \(y=(x - 5)/(x - 4)\), the limit as \(x\) approaches 1 is \(4/3\) and the limit as \(x\) approaches \(\infty\) is \(-\infty\).

Step 8 ：From the observation, we can see that option B is the only one that has a vertical asymptote at \(x=1\) (since the limit as \(x\) approaches 1 is \(-\infty\)) and a horizontal asymptote at \(y=4\) (since the limit as \(x\) approaches \(\infty\) is 4). Therefore, option B is the correct answer.

Step 9 ：Final Answer: \(\boxed{B. \ y=\frac{4 x-5}{x-1}}\)