Determining Limits Using Algebraic Properties of Limits AB & BC

When $\lim_{x\to a} f(x)$ and $\lim_{x\to a} g(x)$ both exist, limits distribute over sums, differences, products, constant multiples, powers, and roots, and over quotients as long as the denominator's limit is not zero. Applying these repeatedly shows that for polynomials, and for rational functions where the denominator does not vanish, the limit is just $f(a)$. That is why direct substitution is valid: it is a theorem about continuous functions, not a shortcut.

Two errors dominate. The first is applying a law whose hypotheses fail, most often the quotient law when the denominator's limit is zero, or splitting an expression such as $\frac{\sin x}{x}$ whose limit exists only as a whole. The laws also do not run backwards: two functions with no limits can sum to one that has a limit. The second is substituting and reporting whatever appears without classifying it, so a $\frac{0}{0}$ gets called "does not exist" when it is an unfinished calculation, and a nonzero over zero gets called 0 when it signals an asymptote.