I think I get how L'Hospital's rule works, at an intuitive level based on the ratios of the derivatives being equivalent to the ratios of the functions and therefore the limit at a point where they would otherwise evaluate to 0/0, so I am happy there. However while learning about it I see something odd in certain explanations, like here:
https://youtu.be/Hu0z-sFfF8Y?t=334
and here
https://youtu.be/kfF40MiS7zA?t=881
If the derivatives are known, then this further criteria or observation of functions "looking like a line" or "being straight" when you "zoom in" doesn't really mean anything here does it? Either you have the derivatives where f(x)/g(x) = 0/0 or you don't. And if you do, L'Hopital's rule applies.
Since we are not using infinitesimals here, then "zooming in" seems like a really sloppy idea here. epsilon delta is at work and this idea of straightness or line-like behavior is meaningless if the derivative is known.
right?
Best Answer
"The function looks like a line when we zoom in" is a hand-wavey, naively intuitive way of saying the function is differentiable at that point. That's all there is to it.