I am working with three functions right now, say $f, g, h$, and I know that
$$\lim_{x \to \infty} \frac{f(x) + g(x)}{h(x) + 1} = 1$$
Furthermore, I know that $$\lim_{x \to \infty} f(x) = 0 \ \ \ \text{ and } \ \ \ \lim_{x \to \infty} h(x) = 0$$
Can I then conclude that $$\lim_{x \to \infty} g(x) = 1 $$ or must I also know that $g$ is convergent before I can apply linearity of limits?
I would love the answer to be yes, and feel as though it should be on an intuitive level, but have been burned in the past by applying linearity where it doesn't belong.
The answer is yes. Let $$ \frac{f(x) + g(x)}{h(x) + 1} = 1+\epsilon(x), $$ where $\lim_{x\to\infty}\epsilon(x)=0$. Since $\lim_{x\to\infty}h(x)=0$, we know that $h(x)+1\ne0$ for $x$ large enough, say $x>R$. Then $$ g(x)=(1+\epsilon(x))\,(h(x)+1)-f(x),\quad x>R. $$