A limit by factoring

Idea. The expression $\frac00$ tells us nothing: it is an *indeterminate form*. We must rewrite the quotient so that the troublesome factor $x-2$ disappears.

Computation. Since $x^2-4=(x-2)(x+2)$, for every $x\neq2$,
$$\frac{x^2-4}{x-2}=x+2.$$
The limit as $x\to2$ only looks at values of $x$ close to $2$ but different from $2$, so we may use this simplification. Therefore the limit is $2+2=4$.

Remark. The function is not even defined at $x=2$, and yet the limit exists. A limit describes what happens *near* a point, not *at* the point.