Most limits ask what happens as x creeps toward a number. A limit at infinity asks something different: what does f(x) head toward as x grows without bound — marching off forever to the right (or left)?

Plug in bigger and bigger inputs — 10, 100, 1000 — and \tfrac{1}{x} shrinks to 0.1, 0.01, 0.001. The curve flattens out and presses closer and closer to the x-axis, never quite touching it.

When f(x) \to L as x \to \pm\infty, the horizontal line y = L is a horizontal asymptote — the height the graph hugs out at the far ends.

Asymptotes of a rational function

For a fraction of polynomials, the trick is to divide top and bottom by the highest power of x and see what survives. Take

The small \tfrac{1}{x} and \tfrac{3}{x} terms vanish, leaving 2. So y = 2 is a horizontal asymptote. A vertical asymptote is different: it sits where the denominator hits zero — here at x = -3 — and the function blows up toward \pm\infty.

Watch the curve flatten

Here is y = \dfrac{2x + 1}{x + 3} drawn against its horizontal asymptote y = 2 (dashed). As x runs to the right, the solid curve rises and presses ever closer to the dashed line — that is the limit at infinity made visible.