Evaluating a difference quotient limit by expansion and factoring

Expand the numerator using $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$. Substituting this in, the numerator becomes $x^3 + 3x^2h + 3xh^2 + h^3 - x^3 = 3x^2h + 3xh^2 + h^3$, so the $x^3$ terms cancel out.

Factor an $h$ out of the remaining numerator to get $h(3x^2 + 3xh + h^2)$, and divide by the $h$ in the denominator, which is valid since $h \neq 0$ as $h \to 0$. This leaves $3x^2 + 3xh + h^2$, and now direct substitution of $h = 0$ is valid, giving the limit $3x^2$.