$\lim_{x\to\infty}{f(x)}=\lim_{x\to\infty}{g(x)}\Rightarrow\lim_{x\rightarrow\infty}{\frac{f(x)}{2^x}}=\lim_{x\rightarrow\infty}{\frac{g(x)}{2^x}}$? Is this statement true? Why?
$$\lim_{x \rightarrow \infty}{f(x)} =\lim_{x \rightarrow \infty}{g(x)} \quad  \Rightarrow\quad \lim_{x \rightarrow \infty}{\frac{f(x)}{2^x}}=\lim_{x \rightarrow \infty}{\frac{g(x)}{2^x}}$$

The answer is definitely no. Think of 2 functions which tends to $\infty$ as $x\to\infty$...
