To find horizontal asymptotes in rational functions, we look at what happens to the function when ( x ) gets really big or really small.

A rational function usually looks like this:

$f(x) = \frac{P(x)}{Q(x)}$

Here, ( P(x) ) and ( Q(x) ) are polynomials, which are just a type of math expression.

Steps to Find Horizontal Asymptotes:

1. Compare Degrees:

   • First, check the degrees. The degree is the highest exponent (or power) in the polynomial.

2. Different Cases:

   • Case 1: Degree of ( P(x) ) is Less than Degree of ( Q(x) )

     • The horizontal asymptote is ( y = 0 ).
     • For example, if we have ( f(x) = \frac{2x}{x^2 + 1} ), as ( x ) gets really big or really small, ( f(x) ) gets closer to 0.

   • Case 2: Degree of ( P(x) ) is Equal to Degree of ( Q(x) )

     • The horizontal asymptote is ( y = \frac{a}{b} ). Here, ( a ) is the leading coefficient of ( P(x) ) and ( b ) is the leading coefficient of ( Q(x) ).
     • For example, in ( f(x) = \frac{3x^2 + 5}{2x^2 + 4} ), the asymptote is ( y = \frac{3}{2} ).

   • Case 3: Degree of ( P(x) ) is Greater than Degree of ( Q(x) )

     • There is no horizontal asymptote, but there might be an oblique asymptote!
     • For example, in ( f(x) = \frac{x^3 + 1}{x + 2} ), as ( x ) gets really big or really small, the graph just keeps rising without limit.

By understanding these cases, we can predict how rational functions behave at the ends!