Horizontal Asymptotes (end behavior, symmetry)

If the degree of the numerator of a rational function is less than the degree of the denominator, the function has a horizontal asymptote. Use division by the highest power of x to determine the equation of the horizontal asymptote.

If the degree of the numerator is one more than the denominator, there is a slant asymptote. Use long division to find the equation of the slant asymptote.

If the degree of the numerator is two or more than the denominator, the end behavior of the function does not follow a linear asymptote. Instead, it is similar to that of a polynomial of the same degree as the difference between numerator and denominator.

What is the end behavior of each of these functions? If there is a horizontal or slant asymptote, what is its equation?

1. f(x) = (2x + 1) / (x – 4)

2. f(x) = (x^{2} + 3) / (x – 1)

3. f(x) = (3x^{4} – x^{3} + 5) / (x^{2} + 7)

4. f(x) = 17 / (x^{2} + 4x – 5)

5. f(x) = (6x^{2} + 1) / (3x^{2} – 1)