If $\lim_{x \to c}[f(x) + g(x)]$ exists, then both $\lim_{x \to c} f(x)$ and $\lim_{x \to c} g(x)$ must individually exist.

If $\lim_{x \to c} f(x) = 0$ and $|g(x)| \leq M$ for all $x$ near $c$ (some constant $M > 0$), then $\lim_{x \to c} f(x)g(x) = 0$.

If $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = +\infty$, then $\lim_{x \to c} f(x)g(x) = 0$.

If $\lim_{x \to c}[f(x) \cdot g(x)]$ exists and $\lim_{x \to c} f(x) = L \neq 0$, then $\lim_{x \to c} g(x)$ exists.