First, regarding the simple way to find the horizontal asymptote.
The location of the horizontal asymptote depends on the function. For example, the function f(x) = 1/x has a horizontal asymptote at y=0.
The step-by-step answer
1. Identify the function's degree (highest power of x) in the numerator and the denominator.
2. Compare the degrees of the numerator and denominator:
a) If the degree of the numerator is less than that of the denominator, the horizontal asymptote is y=0.
b) If the degrees are equal, divide the leading coefficients to find the horizontal asymptote: y=(leading coefficient of numerator)/(leading coefficient of the denominator).
c) If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
For example, consider the function f(x) = (2x^2 + 3)/(x^2 - 5x + 6). The degrees of the numerator and denominator are both 2. Divide the leading coefficients: y = 2/1. So, the horizontal asymptote is y=2.