Question

explain l-hospital rule with a suitable example....

Answer 1

L’Hospital’s Rule tells us that if we have an indeterminate form 0/0 or


all we need to do is differentiate the numerator and differentiate the denominator and then take the limit.

Before proceeding with examples let me address the spelling of “L’Hospital”. The more modern spelling is “L’Hôpital”. However, when I first learned Calculus my teacher used the spelling that I use in these notes and the first text book that I taught Calculus out of also used the spelling that I use here.

Also, as noted on the Wikipedia page for L’Hospital's Rule,

“In the 17th and 18th centuries, the name was commonly spelled "l'Hospital", and he himeself spelled his name that way. However, French spellings have been altered: the silent 's' has been removed and replaced with the circumflex over the preceding vowel. The former spelling is still used in English where there is no circumflex.”

So, the spelling that I’ve used here is an acceptable spelling of his name, albeit not the modern spelling, and because I’m used to spelling it as “L’Hospital” that is the spelling that I’m going to use in these notes.



Example 2 Evaluate the following limit.



Solution
Note that we really do need to do the right-hand limit here. We know that the natural logarithm is only defined for positive x and so this is the only limit that makes any sense.

Now, in the limit, we get the indeterminate form


. L’Hospital’s Rule won’t work on products, it only works on quotients. However, we can turn this into a fraction if we rewrite things a little.



The function is the same, just rewritten, and the limit is now in the form


and we can now use L’Hospital’s Rule.




Now, this is a mess, but it cleans up nicely.



we can always write a product of functions as a quotient by doing one of the following.




Using these two facts will allow us to turn any limit in the form


into a limit in the form 0/0 or


. Which one of these two we get after doing the rewrite will depend upon which fact we used to do the rewrite. One of the rewrites will give 0/0 and the other will give


. It all depends on which function stays in the numerator and which gets moved down to the denominator.


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