Algebra of Limits and Standard Limits

Limits of complicated expressions are almost never worked out from the informal definition directly — instead we build them from limits of simpler pieces, using the algebra of limits. If $\lim_{x\to a} f(x) = L$ and $\lim_{x\to a} g(x) = M$ (both finite), then:

• Sum/Difference rule: $\lim_{x\to a} [f(x) \pm g(x)] = L \pm M$
• Constant multiple rule: $\lim_{x\to a} [k\,f(x)] = k L$ for any constant $k$
• Product rule: $\lim_{x\to a} [f(x)\,g(x)] = L M$
• Quotient rule: $\lim_{x\to a} \dfrac{f(x)}{g(x)} = \dfrac{L}{M}$, provided $M \neq 0$

Alongside these rules, a handful of standard limits are quoted and used directly (their formal derivation belongs to a more advanced course, but every business mathematics syllabus uses them as known results, exactly as $\sin 30° = \tfrac12$ is used without re-deriving it each time):

$\lim_{x\to 0} \frac{e^{x}-1}{x} = 1 \qquad\qquad \lim_{x\to 0} \frac{\log(1+x)}{x} = 1$ …