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Limits and the indeterminate form 0/0

Why 0/0 is not an answer and how simplification reveals a limit’s behavior.

$0/0$ is not the value of a limit. It signals that both numerator and denominator vanish and their ratio needs closer analysis.

An example

Consider $\lim_{x\to2}\frac{x^2-4}{x-2}.$ Direct substitution gives $0/0$ , but $x^2-4=(x-2)(x+2)$ . For $x\ne2$ the ratio is $x+2$ , so the limit is $4$ .

Why cancellation is valid

A limit concerns values arbitrarily close to the point; the function need not be defined at the point itself. The simplified function agrees with the original everywhere near 2 except at 2.

Strategy

After seeing $0/0$ , try factoring, rationalizing, trigonometric identities, or an appropriate theorem. Do not cancel terms across addition: only common factors cancel.

Frequently asked questions

Can 0/0 equal any number?

As an expression it is undefined. Different limits that formally produce 0/0 can have different values, diverge, or fail to exist.

Must I always use l’Hôpital’s rule?

No. In many elementary problems factoring or rationalizing is more direct and exposes the structure more clearly.