Suppose $f,g: (a,+\infty) \rightarrow \mathbb{R}$ are continuous and differentiable with $f(x) \rightarrow 0$ and $g(x) \rightarrow 0$ as $x \rightarrow \infty$. If $g'(x) \neq 0$ on $(a, +\infty$ and $\frac{f'(x)}{g'(x)} \rightarrow l$ as $x \rightarrow \infty$, then $\lim_{x\rightarrow\infty} \frac{f(x)}{g(x)}=l$.
Prove this theorem by applying L'Hopital's Rule (in the case where $x \rightarrow a$) to $f(1/x)/g(1/x)$. I know that someone has already asked this question
However, I am still confused by the explanation given. Specifically, could someone answer two questions: (1) are we dealing with a direct application of the LHR, i.e. we alter the representation of our functions to allow for a direct application, and (2) why is that
$\lim_{x \rightarrow 0}f(1/x)=\lim_{u\rightarrow \infty}f(u)=0$
