Most things in the world settle down: nudge a dimmer switch and the brightness eases to a steady level, and pinning down where it lands is exactly a limit. But some things refuse to settle — a phone bill that leaps the instant you cross your data cap, an amplifier that screeches toward infinity as it overloads, a flickering tube that never holds still. Those are the moments a limit fails, and this page names the ways it can happen.

A limit is a promise: as x approaches c, the outputs settle on one number L — get close enough to c, and f(x) will be as close to L as you demand. The remarkable thing is how often the promise is kept: every polynomial keeps it everywhere, and so do \sin, \cos, e^x and any sensible combination of them. But it's the failures that force every word of the definition to earn its place — "one" number, "settle", "from both sides". Study how the promise breaks, and you'll understand exactly what it means to keep it.

So this page is a rogues' gallery. Mathematics has booked three repeat offenders, each of whom breaks the promise in their own signature style:

1. The Jump — the two sides of c head to different heights. Each side keeps its half of the promise; they just made different promises.
2. The Blow-up — the outputs abandon every finite target and run off towards \pm\infty.
3. The Wild Oscillation — the sneakiest of the three: the outputs stay politely bounded but never stop moving, sweeping back and forth forever.

We'll take their fingerprints one at a time. And watch out for a fourth character — an impostor: a hole in the graph, where f(c) is missing or misplaced, looks alarming but is not a failure at all. A limit deliberately ignores what happens at c; it only interrogates the approach. Each real villain breaks the rule we met in one-sided limits: a two-sided limit exists only when both sides agree on a single, finite value.

Villain 1: the Jump — the sides disagree

Case file. The cleanest jump in mathematics is the sign-detector

which answers "is x positive or negative?" with a +1 or a -1. Let's take its fingerprints at c = 0, one side at a time.

Step 1 — approach from the right. For x > 0 the absolute value changes nothing: |x| = x, so

Step 2 — approach from the left. For x < 0 the absolute value flips the sign: |x| = -x, so

Step 3 — the verdict. Both one-sided limits exist, and each is a perfectly respectable finite number. But 1 \neq -1: anyone standing just left of 0 swears the function is heading to -1, anyone just right swears it's heading to +1, and both are right. So \lim_{x \to 0} |x|/x does not exist (written DNE).

Note what didn't matter: f(0) is undefined here (division by zero), but that's not the crime. Define f(0) = 7 if you like — the limit still fails, because the limit never looks at c itself. The crime is the disagreement. The chart below shows the same failure for a generic step: the left branch runs to height 2, the right branch to height 0, and no amount of zooming near x = 1 will ever reconcile them.

You can commit — and undo — this crime yourself. Below, L sets the height the left branch approaches at x = 1, and R the height the right branch starts from. Whenever L \neq R the limit at 1 does not exist; the instant L = R the branches fuse and the limit exists, equal to that common value. Existence is exactly agreement, nothing more.

Villain 2: the Blow-up — off to infinity

Case file. Take

near c = 0, and just watch the numbers:

Squaring makes the sign irrelevant, so the left side does exactly the same: f(-0.001) is also a million. Could the outputs be settling on some enormous finite L — a billion, say? No: name any ceiling M you like, and as soon as |x| < 1/\sqrt{M} the outputs have punched through it. They commit to no finite target whatsoever, so the limit does not exist. The graph has a vertical asymptote at x = 0 — the chart below has to give up and leave a gap, because no picture is tall enough.

You will see the notation

and it is genuinely useful — but read it carefully. It is not saying the limit exists and equals a number called infinity. \infty is not a number: you can't have outputs "close to" it. The equation is shorthand for a description of the failure: "this limit does not exist, and it fails in the most orderly way possible — both sides grow without bound, in the same direction". It's a police report, not an address.

The disorderly cousin. Now drop the square:

Coming from the right, x is a small positive number, so 1/x is huge and positive: \lim_{x \to 0^+} 1/x = +\infty. Coming from the left, x is small and negative, so 1/x is huge and negative: \lim_{x \to 0^-} 1/x = -\infty. The two sides don't just fail — they fail in opposite directions. Here even the shorthand "= \infty" is off limits: there's no single direction to report. All we may write is that the limit does not exist.

The same analysis works at any asymptote. For \dfrac{1}{x - a} near x = a, just ask: is the denominator a small positive or a small negative number on this side? From the right, x - a is small positive, so the outputs fly to +\infty; from the left, small negative, so they plunge to -\infty. That one question — the sign of the small denominator — settles every one-sided infinite limit you'll meet at A-level.

Villain 3: the Wild Oscillation — never settling

Case file. The strangest exhibit in the gallery:

Away from zero this is tame. But as x \to 0, the inner value 1/x races off to infinity — so the sine, which completes one full wiggle every time its input advances by 2\pi, wiggles faster and faster. Between x = 0.01 and x = 0.001 alone, 1/x sweeps from 100 to 1000: about 143 complete oscillations in a window a hairsbreadth wide. Every shrinking window around 0 contains infinitely many full waves.

Convicting it, step by step. Suspicion isn't proof — let's demonstrate that no candidate L can work. The trick is to walk towards 0 along two different paths and catch the function telling each path a different story.

Step 1. March towards zero stepping only on the points x_n = \dfrac{1}{2\pi n} for n = 1, 2, 3, \dots These certainly approach 0. At each one, 1/x_n = 2\pi n, and \sin(2\pi n) = 0. So along this path the outputs are 0, 0, 0, \dots — apparently heading to 0.

Step 2. Now march in along x_n = \dfrac{1}{2\pi n + \pi/2} instead — also marching straight to 0. But now 1/x_n = 2\pi n + \pi/2, and \sin\!\left(2\pi n + \tfrac{\pi}{2}\right) = 1 every single time. Along this path the outputs are 1, 1, 1, \dots — apparently heading to 1.

Step 3 — the verdict. If the limit were some number L, every approach to 0 would have to produce outputs closing in on that same L. But we've found approaches locked on 0 and approaches locked on 1 — and you could just as easily hit -1, or any value in [-1,1], each visited infinitely often in every window. No single L can be the destination of them all. The limit does not exist.

Notice this villain's signature: unlike the jump, the one-sided limits don't disagree — neither one-sided limit exists at all. And unlike the blow-up, nothing runs away: the whole crime happens inside the strip -1 \le y \le 1. The chart below is honest about defeat — near the middle the waves crowd together faster than any screen can draw, collapsing into a solid blur. That blur is the failure.

The field guide: interrogating a suspect limit

Faced with "does \lim_{x \to c} f(x) exist?", put the function in the interview room and ask three questions, in this order:

• 1. Do the two sides tell the same story? Compute (or read off) the behaviour as x \to c^- and x \to c^+ separately. Two different finite answers — say 2 from the left and 5 from the right — is the Jump. Case closed: DNE. (And no, the limit is not the average — 3.5 is a value the outputs never even approach.)
• 2. Do the outputs stay bounded? If either side runs off to \pm\infty, that's the Blow-up: DNE. Then describe the failure precisely: same direction on both sides (like 1/x^2) earns the shorthand \lim = +\infty; opposite directions (like 1/x) earns nothing but "DNE". The sign of the small denominator on each side is your witness.
• 3. Do the outputs actually settle? Bounded is not enough — if the values keep sweeping through a whole range forever (the giveaway: something like \sin or \cos of a quantity that blows up), that's the Wild Oscillation: DNE, even though nothing explodes.

Pass all three interrogations and the suspect is innocent: both sides agree on one finite value, and the limit exists — even if f(c) itself is undefined or sitting at the wrong height. That's the impostor from the introduction: a hole. For example, \dfrac{x^2 - 9}{x - 3} is undefined at x = 3, but everywhere else it simplifies to x + 3, so the outputs calmly approach 6 from both sides. The limit exists and equals 6. Undefined at a point is a missing value, not a broken promise.

Watch the discontinuities

A second pass through the gallery: Sal Khan walks the jump and the vertical asymptote in graph after graph — good practice at spotting each villain's silhouette at a distance.