A limit is a promise: as x approaches c, the outputs settle on one number L. Sometimes that promise can't be kept, and we say the limit does not exist (often written DNE). There are three classic ways it fails.

1. A jump — the two sides head to different heights.
2. Blowing up — the outputs run off to \pm\infty.
3. Wild oscillation — the outputs never settle at all.

Each builds on one-sided limits: a two-sided limit exists only when both sides agree on a single, finite value.

1. The jump

We met this in one-sided limits. The function steps from one height to another at x = c: \lim_{x \to c^-} f(x) \ne \lim_{x \to c^+} f(x). Both one-sided limits exist, but they disagree — so the two-sided limit DNE.

2. Blowing up to infinity

Near a vertical asymptote the outputs grow without bound. Take f(x) = \frac{1}{x^2}. As x \to 0 from either side, the values shoot up toward +\infty. There is no finite L to land on, so the limit DNE. (We do sometimes write \lim_{x \to 0} \tfrac{1}{x^2} = +\infty to describe how it fails — but "\infty" is not a number, so the limit still doesn't exist in the ordinary sense.)

3. Endless oscillation

The strangest case: f(x) = \sin\!\left(\frac{1}{x}\right). As x \to 0 the input \tfrac{1}{x} races off to infinity, so the sine wiggles between -1 and 1 faster and faster — infinitely many times in any tiny interval near 0. The outputs never home in on a single value, so the limit DNE.

Watch the discontinuities