Saturday, 3 December 2016

real analysis - Use L'Hopital's rule to show that limxrightarrow+inftyfracf(x)g(x)=ell

Let f:RR, g:RR, be both differentiable. Suppose that lim, that g'(x) ≠ 0 for all x \in \mathbb{R} and \lim_{x \rightarrow +\infty} \frac{f'(x)}{g'(x)} = \ell \in \mathbb{R}. Show that




\lim_{x \rightarrow+\infty} \frac{f(x)}{g(x)}= \ell



I'm immediately thinking L'Hopital's rule, and investigating when x tends to an element of \mathbb{R}. I just learnt this however, how would I go forth to use this (assuming I do actually need to use L'Hopital's rule)?

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