Algebraic Limit Theorem for Functional Limits

Limit Laws (Algebraic Limit Theorem) for Functional Limits

Let $f, g : A \to R$ where $A \subseteq R$ and $c$ is a Limit Point of $A$ . Let $L, M \in R$ . If:

lim_{x \to c} f (x) = L, lim_{x \to c} g (x) = M

then:

$lim_{x \to c} [α f (x)] = α L$ for all $α \in R$ .
$lim_{x \to c} [f (x) + g (x)] = L + M$
$lim_{x \to c} [f (x) g (x)] = L M$
$lim_{x \to c} \frac{f (x)}{g (x)} = \frac{L}{M}$ provided $M \neq 0$ .

Proof

Use the Sequential Criterion For Functional Limits, but now using the Limit Laws (Algebraic Limit Theorem) for sequences for each.

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