Limits · Lesson 3 of 5 · ~6 min

When limits fail

Predict

Three curves each go wrong at one marked point — a hole, a jump, and a vertical blow-up. Which of them still has a limit at that point?

Hold that thought. Press Jump in the instrument and drag probe x toward the dashed line, then try Blow-up — watch what the f(x) readout commits to.

Try

Try itPress Jump and drag probe x slowly across $x = 1$, watching the f(x) stat. Press Blow-up and walk the probe toward $x = 1$ from both sides. Then press Oscillate and step toward $0$ as slowly as the slider allows.

Notice

On Jump, probe $x = 0.95$ reads $f(x) = 1.950$ and $x = 1.05$ reads $2.950$: the sides commit to $2$ and $3$ and never close the gap. On Blow-up, $f(1.1) = 100$ and $f(1.01) = 10000$ — each step closer multiplies the reading. On Oscillate, the curve packs about $143$ full cycles between $x = 0.01$ and $x = 0.001$, so the readout lands somewhere new at every step.

Explain

A limit exists at a point when the outputs on both sides agree on one finite number. The hole from Lesson 1 passes that test: $\sin(x)/x$ is undefined at $0$, yet every nearby output crowds $1$. The jump fails it — left outputs pin $2$, right outputs pin $3$, and no single value satisfies both. The blow-up fails it — outputs pass $100$, then $10000$, then any bound: no finite target. The oscillation fails it — outputs keep sweeping $[-1, 1]$ and never settle.

Note what never mattered: whether $f$ is defined at the point. The hole is undefined at $0$ and has a limit. The jump is defined at $1$ — the solid dot says $f(1) = 3$ — and has none. A limit is a claim about neighbors, not about the point itself:

\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L, \quad L \text{ finite}

Apply

Checkpoint — prove it to yourself

On the Jump curve, the solid dot says $f(1) = 3$. Does $\lim_{x \to 1} f(x)$ exist?

Let $g(x) = 1/(x-2)^2$. Compute the trend as $x \to 2$: $g(2.1) = 100$, $g(2.01) = 10000$. What is $\lim_{x \to 2} g(x)$?

New function: $h(x) = \dfrac{x^2 - 9}{x - 3}$. Note that $h(3)$ is undefined. What is $\lim_{x \to 3} h(x)$?

Key idea

A limit reports where nearby outputs are heading. It fails three ways: the sides disagree (jump), the outputs outgrow every bound (blow-up), or they never settle (oscillation). Whether $f(a)$ is defined is a separate question: a hole has a limit, the jump has a value but none.

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