A curious proof of L'Hospital's rule

The claim about $\lim R'(x)$ is wrong. $R(x)\to R$ does not imply $R'(x)\to 0$. Consider $R(x)=R+x$.

Then again, the claim about $\lim R'(x)$ is unnecessarily strong. It would be sufficient to know that $R'(x)$ is bounded near $x=0$ - but is it?

Finally, the text shows (or rather attempts to show) a too weak claim because the final equation $$\lim_{x \rightarrow 0} g'(x)=R \lim_{x \rightarrow 0} h'(x),$$ is not l'Hôpital's rule. In the setup of the text, l'Hôpital's rule might be formulated as

Let $I$ be an interval with $0\in I$. Let $g,h\colon I\setminus\{0\}\to\mathbb R$ be differentiable. Assume $\lim_{x\to0} g(x)=\lim_{x\to0} h(x)=0$ and that $\lim_{x\to 0}\frac{g'(x)}{h'(x)}$ exists. Then $\lim_{x\to 0}\frac{g(x)}{h(x)}$ exists and $\lim_{x\to 0}\frac{g(x)}{h(x)}=\lim_{x\to 0}\frac{g'(x)}{h'(x)}$.

Note specifically that this does not say anything about the existence of $\lim_{x\to 0} g'(x)$ or $\lim_{x\to 0} h'(x)$.

Answer by Hagen von Eitzen has amply demonstrated the problems with the proposed proof of L'Hospital's rule given in your post. I would like to focus on the "limits of popularization of mathematics" which you refer to at the end of your post.

I would like to quote Richard Feynman here (from "Acknowledgment" of "QED"):

Many “popular” expositions of science achieve apparent simplicity only by describing something different, something considerably distorted from what they claim to be describing. Respect for our subject did not permit us to do this. Through many hours of discussion, we have tried to achieve maximum clarity and simplicity without compromise by distortion of the truth.

The issue is even more important to understand in case of mathematics because it is far more rigorous and abstract. Any popularization of ideas of mathematics must not try to downplay rigor. It is rather very very unfortunate that not only popularizations of mathematics avoid rigor but many standard textbooks on calculus are also guilty of the same crime.

In my opinion it is better to state a result without proof (with common excuses that it is beyond the scope of the book or it will be taught in higher classes) than to supplement it with a wrong/non-rigorous proof. The proof presented in your post should be treated as gibberish and nothing more. I wonder what is the motivation of such book authors like P. Nahin.

I remember an old story from my childhood (13 yrs of age, 8th standard) when I asked a teacher about the "proof of irrationality of $\pi$" and the teacher instead gave me an essay on the history of $\pi$ and some 20 digits of $\pi$. It would have been much better if the teacher had told that irrationality of $\pi$ is on an altogether different level as compared to the irrationality of $\sqrt{2}$ and you would probably learn the proof in higher classes. Frankly I was so pissed off with the essay because I had already studied the same essay on $\pi$ from the 9th standard maths textbook.

However we do have some hope on "popularization of mathematics" by presenting the concepts in language instead of symbols (this requires effort on part of author to write more and also needs more pages so that the book would be costlier) and replacing some contrived or unnecessarily complicated proofs with simpler ones.