Let $f,g$ be two functions defined on $[a, \infty] $ for some $a$ .
Give a counterexample to the following statements:
1) If $ \lim_{x\to \infty } (f(x)-g(x)) = \infty $ , then $ \lim_{x\to \infty } (\frac{f(x)}{g(x)} )= \infty $
2) If $ \lim_{x\to \infty } (\frac{f(x)}{g(x)} )= \infty $, then $ \lim_{x\to \infty } (f(x)-g(x)) = \infty $
3) If $ \lim_{x\to \infty } |\frac{f(x)}{g(x)} |= \infty $, then $ \lim_{x\to \infty } |f(x)-g(x)| = \infty $
Thanks in advance