LN_1_6 - M408D: Calculus II Spring 2005 Lecture Notes 1....

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Unformatted text preview: M408D: Calculus II Spring 2005 Lecture Notes 1. LHOSPITALS RULE 1.1. Limits revisited. You should be convinced by now that limits are what calculus is all about. Indeed, differentiation and integration are both limit processes. Recall the definition of the derivative: f prime ( x ) = lim h f ( x + h )- f ( x ) h . Likewise, if f is continuous on the interval [0,1], we define integraldisplay 1 f ( x ) dx = lim n 1 n n summationdisplay k = 1 f parenleftbigg k n parenrightbigg . As to the evaluation of limits, we have the limit laws [ St, 2.3 ]: lim x a [ f ( x ) g ( x )] = lim x a f ( x ) lim x a g ( x ), lim x a [ f ( x ) g ( x )] = bracketleftbig lim x a f ( x ) bracketrightbigbracketleftbig lim x a g ( x ) bracketrightbig , etc., but these are often insufficient. For example, in M408C you probably had to resort to all sorts of tricks to evaluate limits such as lim x 1 radicallow x + 3- 2 x- 1 , lim x sin x x , lim x (1 + x ) 1/ x . The purpose of this lecture is to provide a more generic approach to such hard problems. 1.2. Indeterminate forms. In general, the term indeterminate form refers to a bad limit. For example, if lim x a f ( x ) = lim x a g ( x ) = 0, we call the limit lim x a [ f ( x )/ g ( x )] an indeterminate form of type 0/0. Notice that by choosing the functions f and g appropriately, we can produce any prescribed answer. For example: if f ( x ) = kx and g ( x ) = x , we have lim x f ( x ) = lim x g ( x ) = 0, lim x f ( x ) g ( x ) = lim x kx x = k , so we can produce any finite limit; if f ( x ) = x and g ( x ) = x 3 , we have lim x f ( x ) = lim x g ( x ) = 0, lim x f ( x ) g ( x ) = lim x x x 3 = lim x 1 x 2 = , so we can produce infinite limits; 1 if f ( x ) = x sin(1/ x ) and g ( x ) = x , we have lim x f ( x ) = lim x g ( x ) = 0, lim x f ( x ) g ( x ) = lim x sin(1/ x ). Since the latter limit does not exist (see [ St, p.74 ]), we see that we can even produce a limit that does not exist at all. There are other indeterminate forms besides 0/0. For example, / , - , 0 , , 1 , and 0 are all indeterminate forms, in the sense that such limits can yield any possible answer. What is going on in each of these limits is a tug of war: one of the terms is trying to force the overall limit to be one thing, while the other term is trying to push in a different (usually in the opposite) direction. Which term wins changes from situation to situation. On the other hand, 0/ , /0, + , , 0 , 1 , and are not indeterminatethey will always yield the same result, because in these the two terms are working togehter, instead of against each other....
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LN_1_6 - M408D: Calculus II Spring 2005 Lecture Notes 1....

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