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). Oct 18 '19 at 1:44
• That'd be great, I'd be very glad if you do :D 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.
– user9464
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. 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 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. Oct 21 '19 at 15:17
• @hardmath: Thanks for your kindness and the bounty.
– user9464
Oct 23 '19 at 18:02