Hi
I need some help on the following questions:
1) Use L'Hopital's rule to obtain:
so if x=0, i get 0 however book's answer says
2) Use L'Hopital's rule to obtain:
The only problem i am having is what does equal to?
P.S
This is NOT L'Hopital's rule! You have taken the limit of the derivative of f which is NOT, in general, the limit of f.
(And even for that you would NOT get "0"- you would get which does not exist. Your f' does not have a limit.)
L'Hopital's rule says that if, at x= a, is the indeterminant form " " then provided the limits on the right exist.
That is, you differentiate the numerator and denominator separately NOT using the "quotient law".
Here, the numerator is and the denominator is which are 0 when x= 0, giving the form so we can apply L'Hopital's rule.
The derivative of is and the derivative of is . Those are both 0 at x= 0 so apply L'Hopital's rule again.
The derivative of is and the derivative of is which are 1 and 6 at x= 0.
The limit is .
Okay, here you did differentiate numerator and denominator separately- although it should not be labeled "f' ".2) Use L'Hopital's rule to obtain:
It doesn't equal anything- is not a number, just shorthand for "gets larger without bound."The only problem i am having is what does equal to?
P.S
Here you have a fixed number, 1, over a denominator that gets larger and larger without bound- think of , , , etc. What is happening to those numbers as the denominator gets larger and larger?