Arthur Mattuck in his Introduction to Analysis book, pg. 220 says, in order to prove L'Hospital's Rule for $\infty/\infty$ case,
Let $L=\lim_{x \to \infty} \frac{f'(x)}{g'(x)}$ and choose $a$ so that
$\frac{f'(x)}{g'(x)} \stackrel{\approx}{\epsilon} L$ for $x>a$.
And prove two approximations below (valid for $x\gg 1$)
$$ \frac{f(x)}{g(x)} \stackrel{\approx}{\epsilon} \frac{f(x)-f(a)}{g(x)-g(a)} \stackrel{\approx}{\epsilon} L $$
His hint is, for the first approximation, that we write
$$ f(x) - f(a) = f(x) [ 1 - f(a)/f(x)] $$
and use limit theorem, for the second, use the Cauchy Mean-value Theorem.
Anyone know how to pursue this proof? Thanks,