SalasSV_10_05

SalasSV_10_05 - 10.5 THE INDETERMINATE FORM(0/0 611 10.5...

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10.5 THE INDETERMINATE FORM (0 / 0) 0) 611 10.5 THE INDETERMINATE FORM (0 / 0) (0 Recall that the quotient rule for evaluating the limit of a quotient f ( x ) / g ( x ) fails when f ( x ) and g ( x ) both tend to 0 (see Section 2.3). We called such limits indeterminates of the form 0 / 0. Here we are concerned with limits of quotients where the numerator and denominator both tend to 0 and elementary methods fail or are difficult to apply. THEOREM 10.5.1 L’H ˆ OPITAL’S RULE (0 / 0)† Suppose that f ( x ) 0 and g ( x ) 0 as x c + , x c , x c , x → ∞ , or x → −∞ . If f ( x ) g ( x ) L , then f ( x ) g ( x ) L . NOTE: This theorem includes the possibility that L = ∞ or −∞ . We will prove the validity of L’H ˆ opital’s rule later in the section. First we demon- strate its usefulness. Example 1 Find lim x π/ 2 cos x π 2 x . SOLUTION As x π/ 2, both the numerator f ( x ) = cos x and the denominator g ( x ) = π 2 x tend to zero, but it is not at all obvious what happens to the quotient f ( x ) g ( x ) = cos x π 2 x . Therefore we test the quotient of derivatives: f ( x ) g ( x ) = sin x 2 = sin x 2 1 2 as x π 2 . It follows from L’H ˆ opital’s rule that cos x π 2 x 1 2 as x π 2 . Wecanexpressallthisonjustonelineusing toindicatethedifferentiationofnumerator and denominator: lim x π/ 2 cos x π 2 x = lim x π/ 2 sin x 2 = lim x π/ 2 sin x 2 = 1 2 . Example 2 Find lim x 0 + x sin x . SOLUTION As x 0 + , both numerator and denominator tend to 0. Since f ( x ) g ( x ) = 1 ( cos x )(1 / [2 x ]) = 2 x cos x 0 1 = 0 as x 0 + , † Named after the Frenchman G. F. A. L’H ˆ opital (1661–1704). The result was actually discovered by John Bernoulli (1667–1748), who communicated the result to L’H ˆ opital in 1694.

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612 CHAPTER 10 SEQUENCES; INDETERMINATE FORMS; IMPROPER INTEGRALS it follows from L’H ˆ opital’s rule that x sin x 0 as x 0 + . In short, we can write lim x 0 + x sin x = lim x 0 + 2 x cos x = 0. Remark There is a tendency to abuse the limit-finding technique given in Theorem 10.5.1. L’H ˆ opital’s rule does not apply in cases where the numerator or the denominator has a finite non-zero limit. For example, lim x 0 x x + cos x = 0 1 = 0, but a blind application of L’H ˆ opital’s rule would lead to lim x 0 x x + cos x = lim x 0 1 1 sin x = 1.
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