tgt

## Tuesday, 26 March 2013

### LOCAL BEHAVIOR of FUNCTIONS PART 2

When we deal with the limits of quantities, very often we have to compare numbers such as:
multiplying two large numbers;
subtracting two large numbers;
multiplying a large number with a small number;
etc...
Though it is easy to check that adding two large numbers is a large number (that is  ), 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
;  ;
Remember that the inverse of a small number is a large number while the inverse of a large number is a small number, that is

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

Clearly we are in full swing to use Hôpital's rule. We have

Since

Therefore we have

Example. Fint the limit