In terms of limits, what does it mean for \( f \) to grow faster than \( g \) as \( x \rightarrow \infty \) ? Choose the correct answer below. A. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=0 \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\infty \) B. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=f(x) \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=g(x) \) O. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=\infty \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0 \) O. \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=g(x) \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=f(x) \)

If \( f \) grows faster than \( g \) as \( x \rightarrow \infty \), it means that the value of \( f(x) \) increases at a rate that significantly outpaces \( g(x) \). This can be mathematically represented as the limit where the ratio of \( g(x) \) to \( f(x) \) approaches zero, or equivalently the ratio of \( f(x) \) to \( g(x) \) approaches infinity. This means that as \( x \) gets larger and larger, \( f(x) \) not only supersedes \( g(x) \), but does so by a growing margin. So, the correct answer is A: \( \lim _{x \rightarrow \infty} \frac{g(x)}{f(x)}=0 \) or \( \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\infty \).