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!

Definitely not. Suppose $f$ and $g$ have limits at infinity and that their ratio is nonzero, then if $\lim_{x\rightarrow\infty} f(x) = -c1$, then the equality breaks down. A similar argument works for $g$.
