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

When is $\lim_{x\to\infty} f(x)\rightarrow g(x)$ true?
If I substitute it with $\lim_{x\to\infty} f(x)-g(x) \rightarrow 0$ then $\lim_{x\to\infty} f(x)/2^{x} \rightarrow g(x)/2^{x}$ is trivially true. Because you can't really say $\infty=\infty$, just add the condition that both $f$ and $g$ converge at $\infty$ to satisfy $\lim_{x\to\infty} f(x)/2^{x} = \lim_{x\to\infty} g(x)/2^{x}$.
