Algebra of Limits: Sum, Difference, Product and Quotient

If $\displaystyle\lim_{x\to a} f(x) = L$ and $\displaystyle\lim_{x\to a} g(x) = M$ both exist, the following rules — together called the algebra of limits — let us break a complicated limit into simpler pieces:

A direct corollary of the product rule (taking $g(x) = k$, a constant) is the constant-multiple rule: $\displaystyle\lim_{x\to a} \big[k\cdot f(x)\big] = k\cdot L$. Together, the sum, difference, and constant-multiple rules mean that the limit of any polynomial can always be found by direct substitution — $\displaystyle\lim_{x\to a} p(x) = p(a)$ for any polynomial $p(x)$ — since a polynomial is built entirely from sums, differences, and constant multiples of powers of $x$.

The quotient rule's condition $M\neq0$ is essential: if the denominator's limit is zero, direct substitution fails and the standard formulae of §4 (built precisely for this $\tfrac{0}{0}$ situation) are needed instead.