16. [-/2 Points] DETAILS MY NOTES SCALCET9 2.6.051. Find the horizontal and vertical asymptotes of the curve. You may want to use a graphing calculator (or computer) to check your work by graphing the curve and estimating the asymptotes. (Enter your answers as comma-separated lists. If an answer does not exist, enter DNE.) \[ y=\frac{x^{3}-x}{x^{2}-5 x+4} \] \( x= \) \( y= \)

We begin with the function   y = (x³ – x)/(x² – 5x + 4). Step 1. Factor the numerator and denominator.  • The numerator factors as x³ – x = x(x² – 1) = x(x – 1)(x + 1).  • The denominator factors as x² – 5x + 4 = (x – 1)(x – 4). Thus, the function can be written as   y = [x(x – 1)(x + 1)]/[(x – 1)(x – 4)]. Step 2. Cancel the common factor. The factor (x – 1) cancels, provided x ≠ 1 (this leaves a hole at x = 1):   y = [x(x + 1)]/(x – 4). Step 3. Find the vertical asymptote. Vertical asymptotes occur where the denominator is zero (and the factor does not cancel). The denominator x – 4 equals zero when x = 4, so there is a vertical asymptote at   x = 4. Step 4. Find the horizontal asymptote. For the simplified function, y = [x(x + 1)]/(x – 4), note that the numerator is a quadratic polynomial (degree 2) and the denominator is linear (degree 1). Since the degree of the numerator exceeds the degree of the denominator, there is no horizontal asymptote. (There is, however, an oblique asymptote, which you may find by polynomial division. But the question asks only for horizontal and vertical asymptotes.) Final Answers:   Vertical asymptote: x = 4   Horizontal asymptote: DNE