Why does l'hopital's rule only work for indeterminate forms?

## 3 Answers

It's more like it was proven to work for indetermine forms $0/0$ and, because of the well known property that *number*/0 = *infninity* (true only in limits), you can easily manipulate your function into $inf/inf$, or even if you have indeterminate form of $inf/0$ or $0/inf$, you can still make it to $0/0$ or $inf/inf$

EDIT: your question is not fully correct as well:

>Why does l'hopital's rule only work for **indeterminate forms**?

There are indeterminate forms which are different than $0/0$, $inf/inf$, $0/inf$ and $inf/0$ and L'hopital's rule doesn't work for them. For example: $0\^0$, $inf-inf$, $1\^inf$, $inf\^0$, $0.inf$

EDIT 2: I made a little mistake, $0/inf$ is not indeterminate at all, you don't even need L'hopital's rule for this one, it's obviously 0. $inf/0$ isn't indeterminate too, it's still inf.
An indeterminate form is what you get when you blindly apply quotient rule when it's not applicable. It's not a mathematical object, nothing equal it, and it can't equal anything. It does not exists as mathematical object, it's just symbol on paper.

Strictly speaking, you can't even write lim f(x)/g(x)=lim f(x)/lim g(x)=0/0, but instead you have to check if you can use quotient rule first before the first equality. However, such writings are accepted, because the way you check if quotient rule is applicable or not is by....taking the limits of the numerator and denominator, which is the same thing you do when you apply it anyway. So for convenient sake, you are allowed to blindly apply quotient rule, get an indeterminate form, then backtrack and try a different method.

L'hopital only works for indeterminate form because you're basically working down on your order of approximation. Here is an analogy. If you know that earth population is about 8 billions people and China has about 1 billion, then you can say China is 1/8 of the earth's population, even though you know that "8 billion" and "1 billion" are inexact numbers; the extra figures would be pointless because the answer is dominated by the largest magnitude. But if you know that your house has about 0 billions people, and your street has about 0 billions people, then you cannot take their ratio; at this point you need more significant figures. This is what happen when you use L'hopital, each time you use it it extracts out the next order of error.
Because if we have the limit of f(x)/g(x) as x -> a and both f and g go to zero, L'Hopital says that we should use f(x) &asymp; f'(a)(x-a) and g(x) &asymp; g'(a)(x-a), so that f(x)/g(x) &asymp; f'(a)/g'(a).

But if, say, f(a) is not zero, this doesn't work, because you get constant terms. The ratio of two functions is equal to the ratio of their derivatives only when both functions approach zero (or infinity).

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