Final answer:
The horizontal asymptote for the function f(x) = 5x³+6 / x²-9x+19 is y = 5x.
Step-by-step explanation:
To find the horizontal asymptote for the function f(x) = 5x³+6 / x²-9x+19, we need to examine the behavior of the function as x approaches positive infinity and negative infinity.
As x approaches positive infinity, the highest degree term in the numerator and denominator, which are both x³, dominate the function. Therefore, the horizontal asymptote is y = 5x³ / x² = 5x.
Similarly, as x approaches negative infinity, the highest degree term in the numerator and denominator still dominate the function, and the horizontal asymptote is y = 5x³ / x² = 5x.