Indeterminate Quotient Forms

Indeterminate Quotient Form tex2html_wrap_inline223

May be the most natural indeterminate form is the quotient of two small numbers or tex2html_wrap_inline223 . Equivalently another natural indeterminate form is the quotient of two large numbers or tex2html_wrap_inline227 . 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 tex2html_wrap_inline229 . 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

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Then we have

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Next we take the ratio function tex2html_wrap_inline241 . Do any needed algebra and then find its limit. Hôpital's rule states that if

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then we have

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Remark. Note that if

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then you can use Hôpital's rule for the ratio function tex2html_wrap_inline241 , by looking for

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In other words, there is no limit where to stop.

Example. Find the limit

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Answer. We have tex2html_wrap_inline255 . Hence

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

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Since

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Therefore we have

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Example. Fint the limit

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Answer. We have

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Hence we can use Hôpital's rule. Since tex2html_wrap_inline269 and tex2html_wrap_inline271 , we have

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So it is clear that we need to use Hôpital's rule another time. But since we proved in the example above

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we conclude that

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Therefore, we have

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Remark. The above examples have a wonderful implication. Indeed, the first example implies that when tex2html_wrap_inline281 then tex2html_wrap_inline283 . The second example implies that when tex2html_wrap_inline281 then tex2html_wrap_inline287 .

Example. Fint the limit

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Answer. Set tex2html_wrap_inline291 and tex2html_wrap_inline293 . We have f(0) = g(0) = 0. So we have all assumptions satisfied to use Hôpital's rule. We have

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Clearly we have

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So we use Hôpital's rule again. Set tex2html_wrap_inline301 and tex2html_wrap_inline303 . Then we have

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Again we have

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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

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Using Taylor Polynomials, we get around a (that is tex2html_wrap_inline319 )

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and

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

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and

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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

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and

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when tex2html_wrap_inline319 . This clearly implies

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

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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

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Answer. First we consider the basic functions which generate the functions involved in this limit, that is tex2html_wrap_inline357 and tex2html_wrap_inline359 . Next we write the Taylor Polynomials of these functions

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and

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

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and

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Hence we have

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Therefore, we have

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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

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Answer. We have

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and

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So we can use Hôpital's rule but we will use Taylor polynomial's technique instead. The basic functions involved are tex2html_wrap_inline359 and tex2html_wrap_inline381 . Taylor Polynomials of these functions are

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and

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Hence we have

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and

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Therefore we have

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which implies

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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