Why is lhopitals rule necessary

L'Hospital's rule is the definitive way to simplify evaluation of limits. It does not directly evaluate limits, but only simplifies evaluation if used appropriately. No, it is not so clear why this would help, either, but we'll see in examples.

This problem illustrates the possibility as well as necessity of rearranging a limit to make it be a ratio of things, in order to legitimately apply L'Hospital's rule. When I was studying calculus for the first time, this was the explanation given to me, and now I want to pass it along to you.

Suppose we are taking the limit of a function as x approaches infinity and whose numerator and denominator subsequently approach infinity as well. Is the numerator rapidly approaching infinity while the denominator is going to infinity more slowly? Or is the denominator speeding toward infinity, and the numerator is lagging behind?

In other words, who is dominating the behavior of the overall limit? Is it the numerator or denominator? Hmm, if only there were a way to observe their individual rate of change, then we can see which function is the dominant function.

Some other types are,. The topic of this section is how to deal with these kinds of limits. As already pointed out we do know how to deal with some kinds of indeterminate forms already. For the two limits above we work them as follows. Notice as well that none of the competing interests or rules in these cases won out!

That is often the case. 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. 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. Now we have a small problem. We know how to deal with these kinds of limits.

However, there are many more indeterminate forms out there as we saw earlier. Note that we really do need to do the right-hand limit here.