When we deal with the limits of quantities, very often we have to compare numbers such as:
- adding two large numbers;
- multiplying two large numbers;
- subtracting two large numbers;
- multiplying a large number with a small number;
- etc...
), it is absolutely not clear what happened if we subtract a large number from another large number. We say that we have an indeterminate form. May be one of the most important indeterminate form is the quotient of two small (or large) numbers. Recall that a number close to 0, will be called a small number, while a number close to 
will be called a large number. Note that 
is a very large number which happens to be negative. Many are confused about this point since they believe that is the smallest "number" among the real numbers. Again large and small here is to be understood in terms of quantities while the set of real numbers has a natural order which is not of concerns to us here.Let us give some indeterminate forms which we will take care of in the next pages:
or
;
;
Remark. The inverse of a small number is a large one, this is true sizewise but we do have to work little harder to find out about the sign of the large number (
is a positive large number while is a negative large number). This is the only time when we have to find out about the 0 whether it is positive or negative so we can say something about it inverse. We will write 0+ to designate a positive small number while 0- will designate a negative small number.Indeterminate Quotient Form
May be the most natural indeterminate form is the quotient of two small numbers or
. Equivalently another natural indeterminate form is the quotient of two large numbers or 
. In both cases, it is very easy to convince oneself that nothing can be said, in other words we have no conclusion. It is very common to see students claiming 
. We hope this page will convince some that it is not the case.Hôpital's Rule: Though this rule was named after Hôpital, it is Bernoulli who did discover it in the early 1690s. This rule answers partially the problem stated above. Indeed, let f(x) and g(x) be two functions defined around the point a such that
Then we have
Next we take the ratio function
. Do any needed algebra and then find its limit. Hôpital's rule states that if
then we have
Remark. Note that if
then you can use Hôpital's rule for the ratio function
, by looking for
In other words, there is no limit where to stop.
Example. Find the limit
Answer. We have
. Hence
Clearly we are in full swing to use Hôpital's rule. We have
Since
Therefore we have
Example. Fint the limit
Answer. We have
Hence we can use Hôpital's rule. Since
and 
, we have
So it is clear that we need to use Hôpital's rule another time. But since we proved in the example above
we conclude that
Therefore, we have
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