Concept preview

What you will be able to do

1. 01Interpret $\lim_{x\to\infty} f(x)$ and $\lim_{x\to-\infty} f(x)$ as the end behaviour of a graph and connect them to horizontal asymptotes.
2. 02Apply the fundamental fact $\lim_{x\to\pm\infty} \tfrac{1}{x^n}=0$ (for $n>0$) as the engine for every limit at infinity.
3. 03Evaluate the limit at infinity of a rational function by comparing degrees or dividing by the highest power of x.
4. 04Evaluate roots at infinity correctly using $\sqrt{x^2}=|x|$, watching the sign as $x\to-\infty$.
5. 05Distinguish a limit AT infinity (a horizontal asymptote) from an infinite limit at a finite point (a vertical asymptote).

So far a limit has asked what a function heads toward as $x$ approaches a finite number. Now we let $x$ run off to the edges of the number line and ask about end behaviour: as $x$ grows without bound to the right we write $\lim_{x\to\infty} f(x)$, and as it decreases without bound to the left, $\lim_{x\to-\infty} f(x)$. The answer, when it is a finite number $L$, means the graph flattens out and hugs the horizontal line $y=L$ far from the origin. That line is a horizontal asymptote.