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I hope here is the right place to ask this question.

Recently I learned to use contour integration and residue theorem in order to evaluate real integrals, while that my teacher and many other internet pages that showed some examples said it's a powerful tool, because they are harder found by using real variable tools, but yet it's possible.

My question is why (from what I saw) most of the teachers and mathematicians dislike L'Hospital rule because it's an overpowered tool while no one has a problem evaluating real integrals using a powerful tool like turning to the complex plane?

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    Not only are the mentioned methods in integration powerful, some integrals can only be solved using these methods. And L'Hospital is a very useful rule , but requires some care when exactly it can be used. – Peter Apr 29 '18 at 18:23
  • Thank you for the comment! How do you know its the only possible way? Could you provide me an example that can be evaluated with contour integration but not with real methods? –  Apr 29 '18 at 18:26
  • There are many (definite) integrals here on the site where an antiderivate does not exist but the definite integral can nevertheless be calculated exactly. I do not remember the precise integrals, but if you search questions with the tag "contour-integration" or similar tags , you will find many examples. Of course, the numerical values could be determined without the powerful tools, but the exact solution cannot be found by integration by parts, substitution, etc. – Peter Apr 29 '18 at 18:29
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    The claims you make depend on taste. I don't believe that most mathematicians dislike L'Hopital. I don't see a reason, not to use contour integrals. – Dietrich Burde Apr 29 '18 at 18:32
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    From many MSE questions that I read, I think l'Hospital's rule is often forbidden in homework questions as a form of student abuse: "we've taught you a better way to solve this problem, but rather than test your knowledge of that, we want you to do it the hard way rather than test your ability to use the better way correctly". – Rob Arthan Apr 29 '18 at 18:33
  • @DietrichBurde I agree, especially in the case of L'Hospital. – Peter Apr 29 '18 at 18:33
  • Maybe I'm mistaken, but I was under the impression that all of the anti-L'Hospital sentiment is from a pedagogical perspective and in the setting of calculus. The problem is that some students will learn it without knowing the conditions in which it can be used, and that they'll use it to subvert learning other techniques for evaluating integrals. Not to mention that the general proof often won't be done until a course in analysis. The limit's without L'hopital tag exists so that beginning calculus students won't complete an exercise using a technique they haven't yet learned. – Kevin Long Apr 29 '18 at 18:34
  • Integrals are hard. Maybe we need all the tools we can find to solve them? – mathreadler Apr 29 '18 at 18:35
  • @KevinLong A good point, for the same reason some teachers do not show the abc-formula or the pq-formula to teach binomial expressions. – Peter Apr 29 '18 at 18:35
  • That was a personal view ofcourse, most of my teachers don't use L'Hospital and forbide it like @RobArthan said, but it's the harder way= better way? –  Apr 29 '18 at 18:37
  • @KevinLong: I don't know if you'd read my (somewhat tongue-in-cheek) comment, but thanks anyway for your balanced explanation of the pedagogical issues. – Rob Arthan Apr 29 '18 at 18:37
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    @Sonkun Check out my rant here: https://math.stackexchange.com/questions/1286699/whats-wrong-with-lhopitals-rule/1286806#1286806 – zhw. Apr 29 '18 at 18:50
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    Why do you assert that most mathematicians dislike L'Hôpital's rule? – copper.hat Apr 29 '18 at 18:51
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    @zhw. Thanks I will/ Copper because all my teachers forbide it, I said above that is a personal opinion from what I saw through school. –  Apr 29 '18 at 18:53
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    Nice answer there! In my opinion it would be better if teacher argument explicitly why they will not allow to use LHR, and then afterwards leave freely to the student to choose which method to apply, even in the exam. I for one, like math because I believe every problem can have more than one solution, even those impossible integrals and that's interesting. –  Apr 29 '18 at 19:04

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