Algebraic_Limit_Theorem_for_Functional_Limits

Definition

Let $f$ and $g$ be functions defined on a common domain $A \subseteq \mathbb{R}$, and assume $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$ for some limit point $c$ of $A$. Then:

1. $\lim_{x \to c} [f(x) + g(x)] = L + M$
2. $\lim_{x \to c} [f(x) \cdot g(x)] = L \cdot M$
3. $\lim_{x \to c} [f(x)/g(x)] = L/M$, provided $M \neq 0$

Proof

The proof follows from the Sequential_Criterion_for_Functional_Limits and the Algebraic_Limit_Theorem for sequences.

For any sequence $(x_n) \to c$ with $x_n \neq c$, we have $f(x_n) \to L$ and $g(x_n) \to M$. By the Algebraic Limit Theorem for sequences, $f(x_n) + g(x_n) \to L + M$.

By the Sequential Criterion, $\lim_{x \to c} [f(x) + g(x)] = L + M$. The other parts follow similarly.