Is it always true that $\lim_{x\to\infty} [f(x)+c_1]/[g(x)+c_2]= \lim_{x\to\infty}f(x)/g(x)$? Is it true that $$\lim\limits_{x\to\infty} \frac{f(x)+c_1}{g(x)+c_2}= \lim\limits_{x\to\infty} \frac{f(x)}{g(x)}?$$ If so, can you prove it? Thanks!

Think of it this way: the equality is true only when $f(x), g(x)$ completely 'wash out' the additive constants at infinity. To be more precise, suppose $f(x), g(x) \rightarrow \infty$. Then
$$
\frac{f(x) + c_1}{g(x) + c_2} = \frac{f(x)}{g(x)} \frac{1 + c_1/f(x)}{1 + c_2 / g(x)}
$$
In the limit as $x \rightarrow \infty$, the right-hand factor goes to 1, and so the left-hand quantity approaches the same limit as $f(x) / g(x)$.
