Let us suppose that there is a function $f(x)$ of the form $g(x)/h(x)$. When limit as $x \to \infty$ cannot be determined outright, one can use l'hopital's rule. Suppose that by performing L'hopital's rule, limit can be determined. But before limit is determined, we obtain $g'(x)/h'(x)$. Suppose that $f(x)$ has another form $i(x)/j(x)$. Again, limit cannot be determined outright, so we perform L'hopital's rule. Then we will get $i'(x)/j'(x)$. Again, let's say this derivative form allows us to calculate limit as $x \to \infty$. Then,
Would $g'(x)/h'(x) = i'(x)/j'(x)$?