SalasSV_10_06 - 616 CHAPTER 10 SEQUENCES INDETERMINATE...

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² 10.6 THE INDETERMINATE FORM ( / / / ) ; OTHER INDETERMINATE FORMS We come now to limits of quotients f ( x ) / g ( x ) where numerator and denominator both tend to . Such limits are called indeterminates of the form / . THEOREM 10.6.1 L’H ˆ OPITAL’S RULE ( / ) Suppose that f ( x ) →±∞ and g ( x ) as x c + , x c , x c , x →∞ or x →−∞ . If f ² ( x ) g ² ( x ) L , then f ( x ) g ( x ) L . NOTE: This theorem includes the possibility that L =∞ or −∞ . While the proof of L’H ˆ opital’s rule in this setting is a little more complicated than it was in the (0 / 0) case,† the application of the rule is much the same. Example 1 Let α be any positive number. Show that (10.6.2) lim x →∞ ln x x α = 0. SOLUTION Both numerator and denominator tend to as x .L’H ˆ opital’s rule gives lim x →∞ ln x x α = lim x →∞ 1 / x α x α 1 = lim x →∞ 1 α x α = 0. ³ † We omit the proof.
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10.6 THE INDETERMINATE FORM ( / / ) ; OTHER INDETERMINATE FORMS ± 617 For example, as x →∞ , ln x x 0.01 0 and ln x x 0.001 0. Example 2 Let k be any positive integer. Show that (10.6.3) lim x →∞ x k e x = 0. SOLUTION Here we differentiate numerator and denominator k times: lim x →∞ x k e x = lim x →∞ kx k 1 e x = lim x →∞ k ( k 1) x k 2 e x =··· = lim x →∞ k ! e x = 0. ± For example, as x , x 100 e x 0 and x 1000 e x 0 Remark The limits (10.6.2) and (10.6.3) tell us that ln x tends to in±nity more slowly than any positive power of x and that e x tends to in±nity faster than any positive integral power of x . In the Exercises you are asked to show that e x tends to in±nity faster than any positive power of x and that any positive power of ln x tends to in±nity more slowly than x . That is, for any positive number α , lim x →∞ x α e x = 0 and lim x →∞ [ln x ] α x = 0. Other comparisons of logarithmic and exponential growth were given in Exercises 23 and 24 of Section 7.6. ± Example 3 Determine the behavior of a n = 2 n n 2 as n . SOLUTION To use the methods of calculus, we investigate lim x →∞ 2 x x 2 . Since both numerator and denominator tend to with x ,wetryL’H ˆ opital’s rule: lim x →∞ 2 x x 2 = lim x →∞ 2 x ln 2 2 x = lim x →∞ 2 x (ln 2) 2 2 =∞ . Therefore the sequence must also diverge to . ± Other Indeterminate Forms: 0 ·∞ , ∞−∞ ,0 0 ,1 , 0 0 0 , 0 0 If f tends to 0 and g tends to (or −∞ )as x approaches some number c (or ±∞ ), then it is not clear what the product f · g will do. For example, as x 1, ( x 1) 3 0, 1 ( x 1) 2 , and ( x 1) 3 · 1 ( x 1) 2 = ( x 1) 0. On the other hand, lim x 1 ± ( x 1) 3 · 1 ( x 1) 4 ² = lim x 1 ± 1 x 1 ² does not exist.
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618 ± CHAPTER 10 SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS A limit of this type is called an indeterminate of the form 0 ·∞ . We handle such indeterminates by writing the product f · g as a quotient f 1 / g or g 1 / f .
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SalasSV_10_06 - 616 CHAPTER 10 SEQUENCES INDETERMINATE...

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