**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

**Remark.** The above examples have a wonderful implication.
Indeed, the first example implies that when then
. The second example implies that when then
.

**Example.** Fint the limit

**Answer.** Set and .
We have *f*(0) = *g*(0) = 0. So we have all assumptions satisfied to
use Hôpital's rule. We have

Clearly we have

So we use Hôpital's rule again. Set and . Then we have

Again we have

In fact another use of Hôpital's rule makes the functions involved
even more complicated. So what do we do in this case? A partial
answer is given but the use of Taylor Polynomials.

**Taylor Polynomial's Technique.** First recall the assumptions of
the original problem: let *f*(*x*) and *g*(*x*) be two functions defined
around the point *a* such that

Using Taylor Polynomials, we get around *a* (that is )

and

where *n* and *m* are natural numbers. Since *f*(*a*) = *g*(*a*) =0, we get

and

But we may have the next derivatives also equal to 0 at *a*. Hence we
are sure that there exist two natural numbers *N* and *M* such that

and

when . This clearly implies

So the job is over. Indeed, it is now clear that the limit

is not a problem and depends on the natural numbers *N* and *M*.

Before we do any example showing the power behind this technique,
recall that one may use all the properties of Taylor Polynomials.

**Example.** Fint the limit

**Answer.** First we consider the basic functions which generate the
functions involved in this limit, that is and .
Next we write the Taylor Polynomials of these functions

and

Note that if more terms are needed, we will come back and put the next terms. Using properties of Taylor Polynomials, we get

and

Hence we have

Therefore, we have

One should appreciate the beauty and power behind this technique in
comparing the above calculations with the ones done under Hôpital's
rule.

**Summary.** If you go back to the above example, the calculations
suggest the following steps to follow when using Taylor Polynomials

**1**- write down the basic functions involved in the limit;
**2**- write down Taylor Polynomials of the basic functions;
**3**- make the appropriate substitutions into the Taylor Polynomials as well as any needed algebraic manipulations;
**4**- put the stuff together and make any necessary algebraic
canceling.

**Example.** Find

**Answer.** We have

and

So we can use Hôpital's rule but we will use Taylor polynomial's technique instead. The basic functions involved are and . Taylor Polynomials of these functions are

and

Hence we have

and

Therefore we have

which implies

**
**

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard.

Tue Dec 3 17:39:00 MST 1996

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