Calculus 1A, Lesson 1: What a limit really is — interactive worksheet

Explain-it-like-I'm-5: imagine walking toward a doorway. A limit doesn't ask "are you at the door?" — it asks "where are you heading?" You can know the destination long before you arrive, and it doesn't even matter if a wall stops you one step short. A limit is the value a function is heading toward as the input creeps up on some number.

Take (x² − 4) / (x − 2) at x = 2. Plug in 2 and you get 0/0 — pure nonsense. But that's only because the formula has a hole there, not because the function misbehaves. Factor the top: (x − 2)(x + 2) / (x − 2) = x + 2 for every x except 2. So as x heads to 2, the function heads to 4 — even though it never actually lands there.

f_{hole}\left(x\right) = \frac{x^{2} - 4}{x - 2}

undefined exactly at x = 2 (a 0/0 hole)

near_{2} = \mathrm{f_{hole}}\left(2.0001\right)

4.0001

sneak up on 2 from just above…

✓ pass

\mathrm{abs}\left(near_{2} - 4\right) < 0.001

the limit is 4 — the value it heads toward, hole or no hole

Real-world hook: your car's speedometer reads your speed at an instant — but speed is distance ÷ time, and an instant has zero time. The speedometer secretly computes a limit: average speed over shorter and shorter intervals, closing in on the instantaneous value.

A second classic, the one behind every wave and oscillation: sin(3x) / x is also 0/0 at x = 0, yet it heads straight for 3 (in general sin(kx)/x → k).

g_{sin}\left(x\right) = \frac{\mathrm{sin}\left(3 \cdot x\right)}{x}

another 0/0 at x = 0

lim_{sin} = \mathrm{g_{sin}}\left(0.000001\right)

3.0000

creep toward 0 → heads to 3

✓ pass

\mathrm{abs}\left(lim_{sin} - 3\right) < 0.001

sin(kx)/x → k as x → 0

Try it yourself: what is the limit of (x² − 9) / (x − 3) as x → 3? (Hint: factor the top the way we did above.)

Show the answer

(x − 3)(x + 3)/(x − 3) = x + 3, which heads to 3 + 3 = 6.

lim_{ans} = 6

✏️ Your turn: replace 0 with the value (x²−9)/(x−3) heads toward as x → 3. We compare it to the function evaluated just shy of 3.

✓ pass

\mathrm{abs}\left(lim_{ans} - \frac{3.0001^{2} - 9}{3.0001 - 3}\right) < 0.01

green when your limit matches where the function is heading