limit at infinity

Instead of asking what a function does near a particular point, sometimes we want the long-run story: as the input keeps growing forever, where does the output settle? Think of a cup of coffee cooling toward room temperature. It never quite reaches it, but as time runs on and on, the temperature levels off at a clear value. A limit at infinity captures exactly this 'where does it end up in the long run' behavior.

We write lim x->infinity f(x) = L to mean that f(x) can be made as close to L as we like by taking x large enough; similarly lim x->-infinity f(x) = L looks at x growing large in the negative direction. Note that infinity is not a number you plug in — it is shorthand for 'x increases without bound.' When such a limit equals a finite number L, the line y = L is a horizontal asymptote: the graph hugs that line out toward the edges. A limit at infinity may instead be infinite (the function grows without bound) or may not exist at all (think of sin(x), which keeps oscillating and never settles).

This idea is how we describe the eventual fate of a process: the steady-state of a circuit, the carrying capacity a population approaches, or which of two algorithms wins for large inputs. A useful rule of thumb for a ratio of polynomials is to compare the highest powers on top and bottom — that comparison decides the end behavior.