End Behavior of Rational Functions

Far out along the x-axis, the small terms in a polynomial stop mattering. If you plug in x = 1,000,000 into 4x³ - x, the -x is nothing compared to 4x^3. So as x → infinity or x → -infinity, a polynomial behaves just like its leading term, and a rational function behaves just like the ratio of the two leading terms. That's why you can ignore everything except the highest-degree pieces top and bottom.

This gives three cases. If the denominator's degree is bigger, the whole fraction shrinks toward 0, because the bottom grows faster than the top. If the degrees are equal, the lower-degree terms wash out and you're left with a constant ratio of leading coefficients, which is the horizontal asymptote on both ends. If the numerator's degree is bigger, the fraction grows without bound in at least one direction, so there's no horizontal asymptote, though there may be a slant one instead.

Square roots in the denominator add a twist: √(x²) is |x|, not x, so you have to split positive and negative infinity and track the sign carefully. This topic is the payoff of everything limits have built up to in this unit: you're using the same idea of a function approaching a value without reaching it, now applied specifically to behavior at the far ends of the graph, and it sets up later work on asymptotes and curve sketching.