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

Question

Isabella Ackerman · Accepted Answer

undefined You typically use L'Hôpital's Rule twice when applying it once doesn't yield a determinate form, such as 0/0 or ∞/∞, and the resulting limit still doesn't provide a definitive answer after the first application. If applying L'Hôpital's Rule once leads to another indeterminate form, then applying it again might help resolve the limit.

Aurora Alvarado · Answer

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!