What’s wrong with l’Hopital’s rule?

Upon looking at yet another question on this site on evaluating a limit explicitly without l’Hopital’s rule, I remembered that one of my professors once said something to the effect that in Europe (where he is from) l’Hopital’s rule isn’t “overused” like it is here in the USA.

My question is, is there some reason not to use l’Hopital’s rule when you have an indeterminate form? I know other techniques but l’Hopital is certainly my go-to. Is there some reason for hostility toward l’Hopital’s rule?

Answer

I can only guess that your professor feels it is an overpowered tool. Consider the limit
$$\lim_{x\to 0} \frac{\sin(x)}{\sin(x)+\tan(x)}$$ There is a lot to be gained for a math student by spending more time on this limit. After all, the numerator is approaching zero, so it seems that the limit could be zero. The denominator is also approaching $0$ so maybe the limit doesn’t exist. Clarity comes when you multiply through by $\frac{\csc(x)}{\csc(x)}$ The limit is $1/2$, which is actually sort of curious! (to me at least when I first learned it) It demonstrates a peculiarity of limits in that they can defy intuition. Mathematicians should experience this regularly. You might lose this sort of thinking experience if you just apply L’Hospital’s rule off the bat. There are also limits like $$\lim_{n \to \infty}\left(1+\frac{x}{n}\right)^n$$ that as a first year calculus student I would not be able to do without L’Hospital’s rule. Even using L’Hospital’s rule I would have found that to be a tricky limit. I’m guessing your professor ultimately feels that L’Hospital’s rule should be used as a last resort, when calculating the limit through other means is just not realistic (at that students current level of ability). Otherwise, the student should really spend time contemplating the limit and try to coax out the answer with some critical thinking. I know a number of my college professors felt this way about students using Tabular integration when doing integration by parts. I was a huge fan of Tabular integration because it made integrals like $$\int x^4\sin(x)\text{d}x$$ a breeze. Eventually I had a professor intentionally write all tests and quizzes so that Tabular integration wouldn’t apply. This forced me to get a lot better at integration by parts, and that skill was truly a blessing when I later started learning about Fourier Series. If you continue on in mathematics you will get to real analysis. You will do limits much more rigorously, and L’Hospital’s rule will probably not be available for use. The intuition you gain now from tangling with limits will benefit you down the road.

Attribution
Source : Link , Question Author : user241336 , Answer Author : graydad

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