Limits at Infinity

End behavior, horizontal asymptotes, and one-sided limits.

The idea

A limit at infinity asks what a function does in the long run, rather than near a particular point. Writing $\lim_{x \to \infty} f(x) = L$ means that the values $f(x)$ settle toward $L$, and stay near it, once $x$ is taken large enough. Nothing is ever substituted for $\infty$: it is not a number but shorthand for taking $x$ beyond every bound. This is the function's end behavior, and when such an $L$ exists the graph flattens against the horizontal line $y = L$, a horizontal asymptote.

For a ratio of polynomials, divide the top and the bottom by the highest power of $x$ that appears. Every term then becomes a constant or a multiple of $1/x^{k}$, and each $1/x^{k}$ tends to $0$, so only the leading terms survive: the lower-order terms never affect the limit.

Infinity enters in one other place. At a vertical asymptote the function, not the input, grows without bound, and there the two sides of the point can behave differently, so the approach carries a label: $x \to a^{+}$ means from above, $x \to a^{-}$ from below.

Ways to work on it

• Walkthrough. Divide by the top power; compare degrees; one-sided limits at asymptotes.
• Practice. End behavior of rational functions.
• Hardest. Evaluate a limit at infinity with a radical in a single step.

Not sure where to start? Take the ten-question placement test.