On a asymptotic equality

Question

Let's say that the following asymptotic equality holds, namely

$$ f(x)\sim g(x).$$

My question, if the above asymptotic equation holds, then the asymptotic equality

$$ \lim_{x\to\infty} \frac{f(x)}{g(x)} =1$$

also holds?

Answer 1

Note that by definition

$$f(x)\sim g(x) \iff \lim_{x\to x_0} \frac{f(x)}{g(x)} =1$$

or also, as an alternative and more general definition useful when $g$ has zeroes in every neighbourhood of $x_0$, that for $x\to x_0$ with $\omega(x)\to 0$

$$f(x)\sim g(x) \iff f(x)-g(x)=o(g(x)) \iff f(x)-g(x)=\omega(x)g(x)$$