Limits at Infinity and Asymptotes
Switch on a phone that has been dead for hours and watch the battery graph climb. It shoots up
fast at first, then eases, then crawls — 90%… 97%… 99%… — forever inching toward
100\% but taking an age to get there. A spreading rumour, a cooling
cup of coffee, a charging capacitor, a car reaching its top speed: astonishingly many things in
the world settle toward a level they approach but never quite reach.
Calculus has a precise name for that settling height — a limit at infinity — and
for the invisible line the graph hugs out at the far end: a horizontal asymptote.
Master these and you can read the long-run fate of a function straight off its formula,
without plotting a single extra point.
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)?
\lim_{x \to \infty} \frac{1}{x} = 0
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
\frac{2x + 1}{x + 3} = \frac{2 + \tfrac{1}{x}}{1 + \tfrac{3}{x}}
\;\xrightarrow[\;x \to \infty\;]{}\; \frac{2 + 0}{1 + 0} = 2.
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.
- \dfrac{1}{x^n} \to 0 as x \to \infty for any n > 0;
- if f(x) \to L at infinity, then y = L is a horizontal asymptote;
- for a rational function, divide top and bottom by the highest power of x (equivalently, compare the degrees of numerator and denominator);
- a vertical asymptote sits wherever the denominator equals zero.
The degree race: three outcomes
Once you divide through, the whole story of a rational function's far-off behaviour comes down to
a race between the top degree and the bottom degree. There are exactly three
finishes.
Example 1 — bottom wins (top degree < bottom degree). Find
\displaystyle\lim_{x\to\infty}\frac{3x+2}{x^2+1}. The highest power below
is x^2, so divide every term by x^2:
\frac{3x+2}{x^2+1}=\frac{\tfrac{3}{x}+\tfrac{2}{x^2}}{1+\tfrac{1}{x^2}}
\;\xrightarrow[\;x\to\infty\;]{}\;\frac{0+0}{1+0}=0.
The denominator grows faster, so the fraction is crushed to 0. Horizontal
asymptote y=0.
Example 2 — a dead heat (equal degrees). Find
\displaystyle\lim_{x\to\infty}\frac{4x^2-x}{2x^2+7}. Divide top and bottom
by x^2:
\frac{4x^2-x}{2x^2+7}=\frac{4-\tfrac{1}{x}}{2+\tfrac{7}{x^2}}
\;\xrightarrow[\;x\to\infty\;]{}\;\frac{4-0}{2+0}=2.
When the degrees tie, the limit is just the ratio of the leading coefficients,
\tfrac{4}{2}=2. Horizontal asymptote y=2.
Example 3 — top wins (top degree > bottom degree). For
\dfrac{x^2+1}{x+1}, dividing through leaves an x
on top that never dies:
\frac{x^2+1}{x+1}=\frac{x+\tfrac{1}{x}}{1+\tfrac{1}{x}}\;\to\;\infty.
There is no horizontal asymptote — the function runs off to infinity. (It does
follow a slanted line, an oblique asymptote, but that is a story for another page.)
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.
Reading a real model
A pond stocks fish and biologists model the population by
P(t) = \frac{500\,t}{t + 4}\quad(t \text{ in years}).
What happens in the long run? Divide top and bottom by t:
P(t)=\frac{500}{1+\tfrac{4}{t}}\;\xrightarrow[\;t\to\infty\;]{}\;\frac{500}{1+0}=500.
The population climbs toward 500 and levels off — that ceiling is the
pond's carrying capacity, the horizontal asymptote P = 500.
The fish never quite reach exactly 500, but for any practical purpose they get as close as you like.
Reading the asymptote off the formula told us the model's whole future in one line.
Two traps snare almost everyone here.
-
Don't just "substitute infinity." Writing
\tfrac{2\cdot\infty+1}{\infty+3}=\tfrac{\infty}{\infty} is meaningless —
\tfrac{\infty}{\infty} is an indeterminate form, not a number.
The divide-through step is essential: divide every term by the highest power of
x in the denominator, and only then do the small
\tfrac{1}{x}, \tfrac{1}{x^2} terms vanish to reveal the real limit.
-
An asymptote is not a fence. A horizontal asymptote describes
end-behaviour only — what the curve does far out to the left and right. In the
middle the graph is free to cross it. The curve
y=\dfrac{x}{x^2+1} has asymptote y=0 yet
slices straight through y=0 at the origin. "Approaches at infinity"
says nothing about the journey there.
Limits at infinity are secretly the mathematics of destiny. Ask "how does this end?" and
you are asking for a limit at infinity:
- a spreading epidemic levelling off as everyone has caught it — a horizontal asymptote;
- a capacitor charging toward the battery's full voltage — approaching, never quite arriving;
- a drug concentration in the blood settling to a steady dose;
- a skydiver hitting terminal velocity, where air resistance balances gravity and
the speed stops rising.
Every one of these is a curve pressing toward a horizontal line it never touches. And that idea —
getting arbitrarily close without ever reaching — is the beating heart of all of calculus.
Making it airtight is exactly what the
epsilon–delta
definition of a limit does: it pins down "as close as you like" with real inequalities,
so "never quite reaches" becomes a theorem rather than a hand-wave.
See it explained