Evaluating the rational function $f$ at $x\=4$ produces a "division by zero" error. However, at $x\=4$, the numerator also evaluates to zero. The indeterminate form $\frac{0}{0}$ is no more a valid real number than is $\frac{2}{0}$, and the function $f$ is not defined at $x\=4$. Since the numerator has a zero at $x\=4$, it must also have as one of its factors the linear factor $x-4$. This is the essence of the Factor and Remainder theorem of elementary algebra. The "factor and cancel" step removes the common factor of $x-4$ from both the numerator and denominator, resulting in an expression that has $x\=4$ in its domain. Evaluating this simplified expression yields 8, and that is what the original function $f$ tends towards as $x$ approaches 4.