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Why is there sometimes such an emphasis on not using L'Hôpital's rule for limits? It seems to me that it is similar to other frequently used calculus methods such as integration by parts or implicit differentiation yet you would never see someone request a solution to an integration problem without using integration by parts.

I know this question isn't strictly related to a maths problem but instead of downvoting, just comment if you think it isn't relevant and I'll remove it.

Alexander51413
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  • I think part of it is that in an introductory course in real analysis, L'Hopital's rule comes much later in the course than the $\varepsilon-\delta$ definition of a limit, so in that context it is important to first learn to approach limits without appealing to L'Hopital's rule. – DMcMor Oct 11 '17 at 20:32
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    You might like my rant here: https://math.stackexchange.com/questions/1286699/whats-wrong-with-lhopitals-rule/1286806#1286806 – zhw. Oct 11 '17 at 21:49
  • The aversion to the use of this tool (from viewpoint of teachers/instructors) is matched only by the desperate desire of students to use it. If plugging doesn't work, just differentiate and plug and you get the answer. The problem with the above mantra is that students never learn the conditions under which the rule can be applied and the conditions under which it will be useful. – Paramanand Singh Oct 12 '17 at 01:20

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The main reason is that L'Hopital's rule uses the values of derivatives at certain points - and those derivatives are themselves limits of type $\frac 0 0$. Using L'Hopital's rule there would make the whole thing a circular argument. Think about how you can prove that $\lim_{x\to 0}\frac{\sin x}x = 1$. Using L'Hopital's rule would be assuming the conclusion: what is the very definition of the derivative of $\sin x$ at $x=0$?

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Another possible motivation, for those that may have taught the subject, is that students often overuse l'Hopital's rule, seeing it as the one method to solve every question (even when other methods or clearer understanding would be better). A dangerous attitude, but the rule is powerful, and sometimes one likes to determine its real domain of use. That is: when do you really NEED to use the rule. I think there is also a challenge or skill testing aspect to it.

Rene Schipperus
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