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To Infinity: Limits at Infinity & the Infinitely Small

What happens to a function out at the far edge of the number line, and how do we tame the slippery '0/0' and 'infinity/infinity' that show up along the way? We close rung 1 by tying the old idea of the infinitely small to the precise modern limit.

Walking toward the far end of the line

So far we have asked what a function does as x approaches some ordinary point a. Now we ask a different question: what does f(x) do as x runs off forever to the right, or off forever to the left? This is a limit at infinity. We are not plugging in 'infinity' as if it were a number — there is no such number. We are describing a trend: as x grows without bound, does f(x) settle toward a single value, or does it keep climbing, or wander?

Take f(x) = 1/x. As x marches off toward larger and larger values — 10, 1000, a million — the output gets tinier and tinier: 0.1, 0.001, 0.000001. It never reaches 0, but it heads there unmistakably. We write lim x->infinity 1/x = 0. The graph hugs the horizontal line y = 0 more and more closely; that line is called a horizontal asymptote, and finding it is exactly asking for the limit at infinity.

Reading end behavior, and the slippery forms

For a ratio of polynomials, end behavior is decided by a tug-of-war between the highest powers on top and bottom. A handy trick: divide every term by the biggest power of x in the denominator, then let the leftover 1/x pieces vanish. Watch it work on f(x) = (3x^2 + 5) / (2x^2 - 7).

lim x->infinity (3x^2 + 5) / (2x^2 - 7)
  = lim (3 + 5/x^2) / (2 - 7/x^2)   [divide top & bottom by x^2]
  = (3 + 0) / (2 - 0)               [each 1/x^2 -> 0]
  = 3/2
Same top and bottom degree, so the limit is the ratio of leading coefficients: 3/2.

Notice what we sidestepped. If you naively let x -> infinity first, the top heads to infinity and the bottom heads to infinity, leaving the meaningless expression infinity/infinity. That is an indeterminate form: the symbols alone do not tell you the answer. The same trouble appears as 0/0 — for example, the difference quotient defining a derivative is exactly a 0/0 situation in disguise. Indeterminate does not mean 'no answer'; it means 'you must look closer.' Here, looking closer (the dividing trick) revealed the answer was a tidy 3/2 all along.

The honest story of the infinitely small

When Newton and Leibniz invented calculus in the 1600s, they reasoned with infinitesimals — quantities they imagined as 'infinitely small': smaller than any positive number you could name, yet somehow not zero. To find a slope, Leibniz would write dy/dx as a ratio of two such infinitely small nudges, dy and dx. The methods produced spectacularly correct answers about planets and tides, which is why people kept using them.

But there was a crack in the logic. Critics asked: in a single calculation, dx behaves like a nonzero number (you divide by it), and then a moment later it is treated as zero (you drop it). Which is it? The philosopher George Berkeley mocked these vanishing quantities as 'the ghosts of departed quantities.' The recipes worked, yet no one could say cleanly why.

  1. The honest fix, worked out by Cauchy and Weierstrass in the 1800s, was to stop talking about a fixed 'infinitely small number' and instead talk about a process of approaching.
  2. Instead of an infinitely small dx, we use a small but ordinary h, do the algebra, and then take the limit as h -> 0 — letting h shrink toward 0 without ever being 0.
  3. This dissolves the paradox: while we compute, h is a genuine nonzero number we may divide by; only at the very end do we ask where the expression is heading as h approaches 0. Nothing is ever both zero and nonzero at the same moment.

Closing rung 1: the tiny and the infinite, made precise

Look at how much one idea has carried. The limit lets us speak of a destination a function approaches without ever arriving (1/x heading to 0). It lets us reach out to infinity and read a function's long-run behavior (the asymptote at 3/2). And it lets us shrink a change toward nothing and still divide by it (the slope as h -> 0). Tiny and infinite are two faces of the same coin, and the limit is what lets us handle both honestly.

That same limit is the thread running through all of rung 1: it is how continuity is defined (the limit equals the value), and in the rungs ahead it is literally the definition of the derivative (a limit of difference quotients) and of the definite integral (a limit of sums). Get comfortable with 'what does this approach?' and the rest of calculus becomes variations on a question you already know how to ask.