Limits

Factor, cancel, plug in — the 0/0 recipe.

The idea

A limit records the value a function approaches, which need not be a value it ever takes.

Definition (Limit).

Writing $\lim_{x \to a} f(x) = L$ means that as $x$ is taken closer and closer to $a$, the values $f(x)$ close in on $L$ — as near $L$ as anyone demands, once $x$ is near enough to $a$.

To illustrate why that is worth asking, consider how fast a car is moving at one instant. The direct measurement, distance divided by time, needs two instants, and at a single one it reads $\tfrac{0}{0}$. Measure over shorter and shorter stretches instead and the answers settle; the limit is the value they settle on.

Notice what the definition never mentions: $f(a)$. Here $x$ stays near $a$, never equal to it, and $f$ need not be defined at $a$ at all — a limit says where a function is heading precisely where its formula gives out. When substituting $x = a$ does work, the limit is that value. The form $\tfrac{0}{0}$ is indeterminate: it carries no information, because a ratio of two quantities both shrinking to zero can settle on any value. The remedy is to rewrite the function — for a ratio, by cancelling the vanishing factor the top and bottom share — into one that agrees with it near $a$ and admits substitution.

Ways to work on it

• Walkthrough. Evaluate a 0/0 limit by factoring and canceling.
• Practice. Two-step: factor, then evaluate.
• Hardest. Evaluate a random 0/0 limit in a single step.

Not sure where to start? Take the ten-question placement test.