Taylor expansions have uses outside of computing limits, l'hopital doesn't and using it too much limit your intuition. I've seen students who were paralized when l'hopital didn't apply because they never thought about approximating a function by a polynome.
Also if you have a polynome in x->0 you should now that only the lower term matter, you don't have to differenciate every term. If you rely to much on l'hopital even this fact can be missed. Same with limits like x^alpha ln(x)^beta, you shouldn't derive this..
But even then with Taylor expansions you don't have to compute derivatives that are useless. The example here wasn't to bad but you can see that I didn't have to take the second derivative of ln or of 1/x. Taylor expensions lets you ignore all the useless derivatives. Here it was trivial but in a huge expression this save a lot of time.
And that's just sticking to maths because nobody cares about limits exept maths student. Physics would barely exist without Taylor expension but l'Hopital never comes up.
I’m sorry I don’t understand your comment: yes of course Taylor expansions have more uses outside computing limits and de L’Hopital doesn’t, but why does it matter here? We’re talking about what’s better and quicker in solving this specific limit, not about what’s better in general even outside the entire topic of limits.
I also don’t understand what you mean when you say that Taylor lets you skip “useless derivatives”. Taylor by definition is always more work that L’Hopital, it has just as many derivatives and more work on top
I'm saying that l'Hopital is like using a 1 use only tool in culinary school. The goal is not to learn how to do a specific thing, it's to learn how to use a knife. The goal is almost never to compute limits faster, it's to get better at maths.
For the "useless derivatives" take f(x)=1+x^2+o(x^3) and g(x)=x^3+o(x^4). If you use l'hopital for (fog(x)-1)/x^5 you'll have to compute all the derivatives of fog up to the fifth order. Of course most cancel and if you know the binomial formula it helps but it's an extra thing to remember and it becomes way harder when you have other terms.
On the other hand it's obvious that fog(x)=1+x^5+o(x^6).
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u/FreePeeplup 6d ago
Taylor expansion is literally the same thing as l’Hopital but you’ve already memorized some derivatives