Question

How do you know when to use L'hospital's rule twice?

Answer 1

As soon as you try substitution and see you're in the form `0/0` or `oo/oo`, you may use l'hospitals. Let's try an example!

`L = lim_(x->0) (e^x- x - 1)/x^2`

Try substitution on this and you will get `0/0`.

`L = lim_(x->0) (e^x - 1)/(2x)`

Now try substitution again to get `L = (e^0 - 1)/(2(0)) = 0/0`. So we may indeed apply l'hospitals once more.

`L = lim_(x-> 0) (e^x)/2`

Now we can evaluate directly and see that the limit is `1/2`.

However there will be times when you may not use l'hospitals more than once. Take the following.

`L = lim_(x->0) (e^x - 1)/x^2`

Try substitution and it'll yield `0/0`.

`L = lim_(x->0) e^x/(2x)`

Now try substitution and you will get `1/0`, which is undefined, therefore the limit DNE. Recall that l'hospitals may only be used when you're of the form `0/0` or `oo/oo`.

Hopefully this helps!