properties of identical two limits by l'hopital's rule 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 $g'(x)/h'(x) = i'(x)/j'(x)$?

If
$$f(x) = \frac{g(x)}{h(x)} = \frac{i(x)}{j(x)}$$
for all sufficiently large $x$ and
$$\lim_{x\to\infty} g(x) = \lim_{x\to\infty} h(x) \in \{0,\pm \infty\}$$
and
$$\lim_{x\to\infty} i(x) = \lim_{x\to\infty} j
(x) \in \{0,\pm \infty\}$$
and
$$\lim_{x\to\infty} \frac{g'(x)}{h'(x)},\qquad \lim_{x\to\infty} \frac{i'(x)}{j'(x)}$$
both exist, then
$$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} \frac{g'(x)}{h'(x)} = \lim_{x\to\infty} \frac{i'(x)}{j'(x)}.$$
