Kinda weird how little I could glean about this one on the webernet. Oh, there were TONS of other examples, but this one is, uhh, too simple? Maybe?
I know that the answer is . I know multiple ways to figure out the answer. My question to you is if I can use l'Hopital's rule on it?
If you use it without factoring first, you get OR which will tend to as . But if you factor first, you get . Now from here you can just say it's and be done with it, but I don't think that holds up to a math teacher (does it?) so IF I continue and get derivatives then it becomes which can be written as .
Intuitively this SEEMS to make sense, but am I missing something? I don't want to demonstrate this only to lose points for not seeing that l'hopital's rule doesn't apply. Can someone show me the light?
No, L'Hopital's rule does NOT apply because the numerator and denominator are not both going to the same thing. L'Hopital's rule only applies if numerator an denominator both go to 0, or both to positive infinity, or negative infinity.
Here, it should be trivial to see that for x very close to 0, from above, the numerator is a very large, negative, number while the denominator is a very small positive number. The quotient is a very large negative number so the limit is negative infinity.