So there's a part of the proof of L'hopital rule given in Pugh's Real Mathematical Analysis that I don't understand. I was going to ask a question, but it turns out that another user had the same confusion, and asked a question earlier(Clarification of L'Hopital Proof Pugh), nobody answered it and it only has a comment, but the comment doesn't address the confusion of the user.

Just like this user, I don't understand why the constraints

\begin{align}|f(t)+g(t)| &< \frac{g(x)^2\epsilon}{4(|f(x)|+|g(x)|)} \\ |g(t)| &< \frac{|g(x)|}{2}.\end{align} imply that $$\frac{g(x)f(t)-f(x)g(t)}{g(x)[g(x)-g(t)]} < \epsilon/2$$

Is it fine if I start a new post?. The other option is to start a bounty, but I have little reputation and I don't want to lose some of my privileges.

• Would you be agreeable for me to post a bounty on your behalf? I think one of the standard reasons will do, such as this Question needs a canonical answer (or words to that effect). – hardmath Oct 18 '19 at 1:44
• That'd be great, I'd be very glad if you do :D – Donlans Donlans Oct 18 '19 at 14:27
• The first constraint has a typo: it is $|f(t)|+|g(t)|$, not $|f(t)+g(t)|$. See the original excerpt here. – Jack Oct 18 '19 at 15:26
• @DonlansDonlans: Have you had a chance to read the Answers that were posted on the previous Questions? If you have some feedback I'd be happy to take it into consideration in awarding the bounty. – hardmath Oct 20 '19 at 14:56
• @hardmath Yes,I have read them, DanielWainfleet's answer and Jack's answer have been the most useful to me. Though if I had to choose, I'd go for Jack's answer, he explained why the constraints imply the inequality that I mentioned on this post,which was what I specifically didn't understand.Thank you very much for starting the bounty :D – Donlans Donlans Oct 21 '19 at 15:15
• @DonlansDonlans: Thanks for the feedback. I'll consider it when awarding the bounty. I've read and upvoted all the answers posted before I asked you to check them out. – hardmath Oct 21 '19 at 15:17
• @hardmath: Thanks for your kindness and the bounty. – Jack Oct 23 '19 at 18:02