Determining Limits Using Algebraic Manipulation AB & BC

A $\frac{0}{0}$ form at $x = a$ means numerator and denominator share a factor of $(x-a)$. Cancelling it yields an expression that agrees with the original everywhere except at $a$ itself, which is enough, because a limit never consults the point. Factoring handles polynomials, multiplying by a conjugate handles radicals, and combining over a common denominator handles complex fractions. After each rewrite, substitute again and reclassify.

Three things go wrong. Substitution gets reported without classification, so an unfinished $\frac{0}{0}$ becomes a final answer. The indeterminate form gets read as a value or as proof the limit fails. And cancellation is applied to a term rather than a factor, as in simplifying $\frac{x^2+3x}{x}$ to $x^2+3$, which produces an expression not equal to the original and therefore a wrong limit rather than an unsimplified one. Sign errors from reversed factors such as $3-x$ against $x-3$ are the common arithmetic slip.