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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:
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adding two large numbers;
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multiplying two large numbers;
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subtracting two large numbers;
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multiplying a large number with a small number;
tex2html_wrap_inline28
etc...
Though it is easy to check that adding two large numbers is a large number (that is tex2html_wrap_inline38 ), 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 tex2html_wrap_inline40will be called a large number. Note that tex2html_wrap_inline42 is a very large number which happens to be negative. Many are confused about this point since they believe that tex2html_wrap_inline42 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:
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tex2html_wrap_inline48
tex2html_wrap_inline28
tex2html_wrap_inline52
tex2html_wrap_inline28
tex2html_wrap_inline56
tex2html_wrap_inline28
tex2html_wrap_inline60 or tex2html_wrap_inline62
tex2html_wrap_inline28
tex2html_wrap_inline66 ; tex2html_wrap_inline68 ; tex2html_wrap_inline70
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
displaymath72

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 ( tex2html_wrap_inline74 is a positive large number while tex2html_wrap_inline42 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 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
displaymath237
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
displaymath245

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
displaymath257
Clearly we are in full swing to use Hôpital's rule. We have
displaymath259
Since
displaymath261
Therefore we have
displaymath263

Example. Fint the limit
displaymath265
Answer. We have
displaymath267
Hence we can use Hôpital's rule. Since tex2html_wrap_inline269 and tex2html_wrap_inline271 , we have
displaymath273
So it is clear that we need to use Hôpital's rule another time. But since we proved in the example above
displaymath263
we conclude that
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Therefore, we have
displaymath279

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