Telescope to Infinity

Calculus 1 · Axiom Academy

Zoom a graph out far enough and a wild curve flattens into a single straight line. That line is a limit at infinity. Up close, many functions look complicated — they rise, dip, wiggle. But the interesting question in calculus is often the long-run one: as x runs off toward infinity, does the curve settle down toward some value , or keep climbing forever? Mount a telescope on the graph and zoom out — the local detail shrinks to nothing and the end behavior reveals itself. Watch the viewport zoom out on f(x) = 3x² / (x² + 1) . Near the origin the curve sweeps up steeply; but as the window widens to show ever-larger x , the curve presses flat against the dashed line y = 3 . That line it never quite reaches is a horizontal asymptote , and the value it homes in on is the limit. The further out you look, the flatter it gets — the curve is being pulled toward its limiting height y = 3. Push x out and watch the gap vanish Drag the slider to push x out toward 100,000. The point marches along the curve and the gap between f(x) and the line y = 3 — shown as the orange bar — shrinks toward zero. "Approaches a limit L" means exactly this: the gap |f(x) − L| can be made as small as you like, just by going far enough out. As x → ∞, the gap → 0 — so we write lim f(x) = 3. The curve never lands on the line, but it gets arbitrarily close. Which value? The degrees decide

This is the written version of the interactive lesson above. See the full Calculus 1 course.