Concept preview

What you will be able to do

1. 01Interpret $\lim_{x\to+\infty} f(x)=L$ and $\lim_{x\to-\infty} f(x)=L$ as statements about a function end behaviour, and connect a finite value $L$ to a horizontal asymptote $y=L$.
2. 02Evaluate limits at infinity of rational functions by comparing the degrees of numerator and denominator (smaller $\to 0$, equal $\to$ ratio of leading coefficients, larger $\to \pm\infty$).
3. 03State the horizontal asymptote(s) of a rational function, or justify that none exists, directly from the degree comparison.

The previous lesson asked what happens as $x$ homes in on a fixed trouble spot. This lesson (AP topic 1.15) flips the question: what happens as $x$ itself marches off toward $+\infty$ or $-\infty$? This is called a limit at infinity, and it captures the long-run, far-out behaviour of a function, its end behaviour. When that far-out behaviour settles toward a single finite height, the graph flattens against a horizontal line called a horizontal asymptote.