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For this equation, $\lim _{x\to \infty }\frac{\left(e^x+e^{-x}\right)}{e^x-e^{-x}}$, I have found the solution:

We divide by $e^x$.

however, when you take into account that it's $\frac \infty \infty$, we should use l'hospital's rule but instead we're diving by $e^x$. Why didn't we use l'hospital's rule given that the equation is infinity/infinity ?

Alan
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COCO21
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  • Attempt to use l'hospital's rule ... you end up right where you started – jameselmore Sep 27 '22 at 18:14
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    have you tried applying L'Hospital's rule? If not, try it and maybe you'll notice something – user1090793 Sep 27 '22 at 18:14
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    @jameselmore Not quite, you get the reciprocal, which does let us say that if the limit exists, it must be a number that is its own multiplicative inverse.... – Alan Sep 27 '22 at 18:17
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    You are never forced to use L'Hôpital's rule; it doesn't make sense to say you "should" use it. It is a tool that is often, but not always, helpful. – Théophile Sep 27 '22 at 18:18
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    To the OP, "can" use L'hospitals is different than "should" use L'hospitals. This one is legal to use it for, so it fits "can". But doing so only lets you conclude that the limit must be 1 IF IT EXISTS, so unless you can prove the limit exists some other way, it doesn't help, thus you "shouldn't – Alan Sep 27 '22 at 18:18
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    @Alan It could also be $-1$. – Théophile Sep 27 '22 at 18:20
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    @Théophile Doh! Okay, I need more caffeine. And jinx on the forced/should part :) – Alan Sep 27 '22 at 18:21
  • Could someone please use l'hopital's rule, that would be pretty beneficial. I feel like this question hasn't been asked before – COCO21 Sep 27 '22 at 18:22
  • It has been asked before, see for example https://math.stackexchange.com/q/1574663/42969 or https://math.stackexchange.com/q/4165886/42969 or https://math.stackexchange.com/a/912694/42969 – Martin R Sep 27 '22 at 18:23
  • Definitely ! Thanks – COCO21 Sep 27 '22 at 18:25
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    The answer to the question "has this been asked before?" is almost always yes, at least in my experience on this site : ) – user1090793 Sep 27 '22 at 18:25
  • Now I can't ask any question :( – COCO21 Sep 28 '22 at 00:27

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